A Comp endium of Problems Complete for P

Preliminary

Decemb er

RCS Revision

Raymond Greenlaw

y

H James Ho over

z

Walter L Ruzzo

Department of Computing Science Department of Computer Science

University of Alb erta University of New Hampshire

Technical Rep ort TR Technical Rep ort TR

Department of Computer Science and Engineering

UniversityofWashington

Technical Rep ort TR

Abstract

This pap er serves two purp oses Firstlyitisanelementary intro duction to the

theory of P completeness the branch of complexity theory that fo cuses on identify

ing the problems in the class P that are hardest in the sense that they app ear to

lack highly parallel solutions That is they do not have parallel solutions using time

p olynomial in the logarithm of the problem size and a p olynomial numb er of pro cessors

unless all problem in P havesuch solutions or equivalently unless P NC Secondly

this pap er is a reference work of P complete problems We present a compilation of

the known P complete problems including several unpublished or new P completeness

results and many op en problems

This is a preliminary version mainly containing the problem list The

latest version of this do cument is available in electronic form by anony

mous ftp from thorhildcsualbertaca as either a compressed

dvi le TRdviZor as a compressed p ostscript le TRpsZ

or from cswashingtonedu as a compressed p ostscript le

truwcsepsZ

c

Raymond Greenlaw H James Ho over and Walter L Ruzzo

Authors address Department of Computer Science University of New Hampshire Durham NH

email address greenlawunhedu

y

Research supp orted by the Natural Sciences and Engineering Research Council of Canada grantOGP

Authors address Department of Computing Science University of Alb erta Edmonton Alb erta

Canada TG H email address hoovercsualbertaca

z

Research supp orted in part by NSF grants ECS CCR CCR and by

NSFDARPA grant CCR A p ortion of this work was p erformed while visiting the Univer

sityofToronto whose hospitality is gratefully acknowledged Authors address DepartmentofCom

puter Science and Engineering FR UniversityofWashington Seattle WA email address

ruzzocswashingtonedu

A Comp endium of Problems Complete for P Preliminary RCS Revision

Additional Problems

The authors would b e pleased to receive suggestions corrections and additional prob

lems Please send new problems bibliographyentries andor copies of the relevantpapers

to Ray Greenlaw at the address on the title page

A Comp endium of Problems Complete for P i

Contents

Part I Background and Theory

Intro duction

Setting the Scene

Feasibility

Completeness

Overview of this Pap er

Parallel Mo dels and Complexity Classes

Parallel Machines

Computation Problems versus Decision Problems

ComplexityClasses

The Complexity Classes P and NC

Reducibili ty

Completeness

Two Basic P complete Problems

The Generic P complete Problem

The Circuit Value Problem

Evidence That NC Do es Not Equal P

General Simulations Are Not Fast

Fast Simulations Are Not General

Natural Approaches Provably Fail

Summary

The Circuit Value Problem

The Circuit Value Problem is P Complete

Restricted Versions of the Circuit Value Problem

Parallel Versions of Sequential Paradigms

Strong P completeness

Approximations to P complete problems

Greedy Algorithms

ii A Comp endium of Problems Complete for P Preliminary RCS Revision

Lexicographically Ordered Greedy Algorithms

Generic Greedy Algorithms

Bo olean Circuits

Bo olean circuits

Uniform Circuit Families

k

The Class NC

k

NC Reducibil ity

Other Notions of Reducibil ity

Random NC

Acknowledgements

Part I I P Complete and Op en Problems

A List of P Complete Problems

A Circuit Complexity

A Circuit Value Problem CVP

A Top ologically Ordered Circuit Value Problem TopCVP

A Monotone Circuit Value Problem MCVP

A Alternating Monotone Fanout CVP AMCVP

A NAND Circuit Value Problem NANDCVP

A Synchronous Alternating Monotone Fanout CVP SAMCVP

A Planar Circuit Value Problem PCVP

A Arithmetic Circuit Value Problem ArithCVP

A MinPlus Circuit Value Problem MinPlusCVP

A Inverting an NC Permutation InvNCPerm

A Graph Theory

A Lexicographically First Maximal Indep endent Set LFMIS

A Lexicographically First Maximal Clique LFMC

A Alternating Graph Accessibility Problem AGAP

A Hierarchical Graph Accessibility Problem HGAP

A Restricted Chromatic Alternating Graph Accessibili ty Problem

RCAGAP

A Lexicographically First Vertex Coloring LFDVC

A Comp endium of Problems Complete for P iii

A High Degree Subgraph HDS

A High Connectivity Subgraph HCS

A Ordered High Degree Vertex Removal OHDVR

A Ordered Low Degree Vertex Removal OLDVR

A Ordered Vertices Remaining OVR

A Nearest Neighbor Traveling Salesman Heuristic NNTSH

A Lexicographically First Maximal Subgraph for LFMS

A Minimum FeedbackVertex Set MFVS

A Edge Maximal Acyclic Subgraph EMAS

A General Graph Closure GGC

A Search Problems

A Lexicographically First Maximal Path LFMP

A Lexicographically First Depth First Search Ordering LFDFS

A BreadthDepth Search BDS

A Stack Breadth First Search SBFS

A Alternating Breadth First Search ABFS

A Combinatorial Optimization and Flow Problems

A Linear Inequalities LI

A Linear Equalities LE

A Linear Programming LP

A Maximum Flow MaxFlow

A Homologous Flow HF

A Lexicographically First Blo cking Flow LFBF

A First Fit Decreasing Bin Packing FFDBP

A General List Scheduling GLS

A Lo cal Optimality

A MAXFLIP Verication MAXFLIPV

A Lo cal Optimality KernighanLin Verication LOKLV

A Unweighted NotAllEqual Clauses SATFLIP UNSATFLIP

A Unweighted MAXCUTSWAP UMS

A Unweighted SATFLIP USATFLIP

A Unweighted SWAP USWAP

A Unweighted STABLE NET USN

iv A Comp endium of Problems Complete for P Preliminary RCS Revision

A Logic

A Unit Resolution UNIT

A Horn Unit Resolution HORN

A Prop ositional Horn Clause Satisability PHCS

A Relaxed Consistent Lab eling RCL

A Generability GEN

A Path Systems PATH

A Unication UNIF

A Logical Query Program LQP

A Formal Languages

A ContextFree Grammar Memb ership CFGmem

A ContextFree Grammar EmptyCFGempty

A ContextFree Grammar Innite CFGinf

A ContextFree Grammar Memb ership CFGmem

A StraightLine Program Memb ership SLPmem

A StraightLine Program Nonempty SLPnonempty

A Lab eled GAP LGAP

A TwoWayDPDA Acceptance DPDA

A Algebraic Problems

A Finite Horizon Markov Decision Pro cess FHMDP

A Discounted Markov Decision Pro cess DMDP

A Average Cost Markov Decision Pro cess ACMDP

A Gaussian Elimination with Partial Pivoting GEPP

A Iterated Mo d IM

A Generalized Word Problem GWP

A Subgroup Containment SC

A Subgroup Equality SE

A Subgroup Finite Index SFI

A Group Indep endence GI

A Group Rank GR

A Group Isomorphism SI

A Group Induced Isomorphism GI I

A Intersection of Cosets IC

A Comp endium of Problems Complete for P v

A Intersection of Subgroups IS

A Group Coset EqualityGCE

A Conjugate Subgroups CS

A Uniform Word Problem for Finitely Presented Algebras UWPFPA

A Triviality Problem for Finitely Presented Algebras TPFPA

A Finitely Generated Subalgebra FGS

A Finiteness Problem for Finitely Presented Algebras FPFPA

A Uniform Word Problem for Lattices UWPL

A Generator Problem for Lattices GPL

A Bo olean Recurrence Equation BRE

A Geometry

A Plane Sweep Triangulation PST

A Oriented Weighted Planar Partitioning OWPP

A Visibili tyLayers VL

A Real Analysis

A Real Analogue to CVP RealCVP

A Fixed Points of Contraction Mappings FPCM

A Inverting a OnetoOne Real Functions IOORF

A Miscellaneous

A General Deadlo ck Detection GDD

A TwoPlayer Game GAME

A Cat Mouse CM

A Longcake LONGCAKE

B Op en Problems

B Algebraic Problems

B Integer Greatest Common Divisor IntegerGCD

B Extended Euclidean Algorithm ExtendedGCD

B Relative Primeness RelPrime

B Mo dular Inversion Mo dInverse

B Mo dular Powering Mo dPower

B Short Vectors SV

B Short Vectors Dimension SV

B Sylow Subgroups SylowSub

vi A Comp endium of Problems Complete for P Preliminary RCS Revision

B TwoVariable Linear Programming TVLP

B Univariate Polynomial Factorization over Q UPFQ

B Graph Theory Problems

B EdgeWeighted Matching EWM

B Bounded Degree Graph Isomorphism BDGI

B Graph Closure GC

B Restricted Lexicographically First Indep endent Set RLFMIS

B Maximal Path MP

B Maximal Indep endent Set Hyp ergraph MISH

B Geometric Problems

B Oriented Weighted Planar Partitioning OWPP

B Limited Reection RayTracing LRRT

B Restricted Plane Sweep Triangulation SWEEP

B Successive Convex Hulls SCH

B Unit Length VisibilityLayers ULVL

B CC Problems

B Comparator Circuit Value Problem CCVP

B Lexicographically First Maximal Matching LFMM

B Stable Marriage SM

B RNC Problems

B Directed or Undirected Depth First Search DFS

B Maximum Flow MaxFlow

B Maximum Matching MM

B Perfect Matching Existence PME

C Undigested Pap ers

References

Problem Index

A Comp endium of Problems Complete for P Preliminary RCS Revision

Acknowledgements

We wish to thank the following for contributing new problems and p ointing us to ap

propriate literature Richard Anderson Paul Beame Anne Condon Steve Co ok Derek

Corneil Joachim von zur Gathen Arvind Gupta John Hershb erger David Johnson Luc

Longpre Mike Luby Pierre McKenzie Andrew Malton Satoru Miyano John Reif Jack

Sno eyink Paul Spirakis Iain Stewart Ashok Subramanian EvaTardos Martin Tompa

and H Venkateswaran

Part I I P Complete Problems

Part I I P Complete and Op en Problems

A List of P Complete Problems

This section contains a list of P complete problems For eachentry wegive a description of

the problem and its input references to source pap ers showing the problem P complete a

hint illustrating the main idea of the completeness reduction and mention related versions

of the problem

Problems marked by are P hard but are not known to b e in P Other problems are

in P although we usually omit the pro ofs that they are

Many of the problems given here were originally shown P complete with resp ect to log

space reduction Any log space computation is immediately in NC by Boro dins simula

NC

tion Bor Thus any problem log space complete for P is also complete for P In

m

NC

most cases the same reduction can b e done in NC ie the problem is complete for

m

P Wehave noted some exceptions to this b elow but have not b een exhaustive

The problems are divided into the following categories circuit complexity graph the

ory search combinatorial optimization and ow lo cal optimality logic formal languages

algebraic geometry real analysis and miscellaneous

A Circuit Complexity

The circuit value problem CVP plays the same role in P completeness theory that satis

abilitydoesinNP completeness theory In this section wegivemanyvariants of CVP that

are P complete and are particularly useful for proving other problems are P complete See

Section of this pap er for more details

A Circuit Value Problem CVP

of a Bo olean circuit plus inputs x x Given An enco ding

n

Problem Do es on input x x output

n

Reference Lad

Hint A pro of is given in Section of this pap er

Remarks For the two input basis of Bo olean functions it is known that CVP is P

complete except when the basis consists solely of or consists solely of and or consists of

any or all of the following xor equivalence andnot GPPar

A Top ologicall y Ordered Circuit Value Problem TopCVP

of a Bo olean circuit plus inputs x x with the additional Given An enco ding

n

assumption that the vertices in the circuit are numb ered and listed in top ological order

A Comp endium of Problems Complete for P Preliminary RCS Revision

Problem Do es on input x x output

n

Reference Folklore

Hint A pro of is given in Theorem Also see the remarks following the pro of of

Theorem

Remarks All of the reductions in Sections and A preserve top ological ordering so the

restrictions of all of these variants of the Circuit Value Problem to top ologically ordered

instances remain P complete

A Monotone Circuit Value Problem MCVP

of a monotone Bo olean circuit that is one constructed solely of Given An enco ding

and and or gates plus inputs x x

n

Problem Do es on input x x output

n

Reference Gol

Hint Reduce CVP to MCVP A pro of is given in Section

A Alternating Monotone Fanout CVP AMCVP

of a monotone Bo olean circuit plus inputs x x Onany Given An enco ding

n

path from an input to an output the gates are required to alternate between or and and

gates Inputs are required to b e connected only to or gates and outputs must come directly

from or gates The circuit is restricted to have fanout exactly two for inputs and internal

gates and to have a distinguished or gate as output

Problem Do es on input x x output

n

Reference Folklore

Hint A pro of is given in Section

Remarks GSSgave a completeness pro of for monotone fanout CVP

A NAND Circuit Value Problem NANDCVP

Given An enco ding of a Bo olean circuit constructed only of nand gates plus inputs

x x The circuit is restricted to havefanouttwo for inputs and nand gates

n

Problem Do es on inputs x x output

n

Reference Folklore

Hint Reduction of AMCVP to NANDCVP A pro of is given in Section

Remarks Any complete basis of gates suces by the obvious simulation of nand gates

in the other basis For example nor gates form a complete basis NOR CVP is dened

analogously to NANDCVP See Post Posforacharacterization of complete bases See

the remarks for Problem A for other bases not necessarily complete for which the

asso ciated circuit value problem is still complete

A Synchronous Alternating Monotone Fanout CVP SAMCVP

of a monotone Bo olean circuit plus inputs x x In addition Given An enco ding

n

to the restrictions of Problem A this version requires the circuit to b e synchronous That

Part I I P Complete Problems

is eachlevel in the circuit can receive its inputs only from gates on the preceding level

Problem Do es on input x x output

n

Reference GHR

Hint A pro of is given in Section The reduction is from AMCVP

A Planar Circuit Value Problem PCVP

Given An enco ding of a planar Bo olean circuit that is one whose graph can b e drawn

in the plane with no edges crossing plus inputs x x

n

Problem Do es on input x x output

n

Reference Gol McC

Hint Reduce CVP to PCVP Lay out the circuit and use crossover circuits to replace

crossing lines with a planar sub circuit A planar xor circuit can b e built from two each

gates a planar crossover circuit can b e built from three planar xor circuits An

m gate CVP instance is emb edded in an m m grid as follows Gate i will b e in cell i i

th

with its value sent along a wire in the i row b oth to the left and to the right Gate is

th

inputs are delivered to it through two wires in the i column with data owing down the

wires from ab ove and up from b elow Let gate is inputs b e the outputs of gates j and k

and supp ose j happ ens to b e less than iIncellj i which happ ens to b e ab ovei i at

the p oint where j s horizontal rightgoing output wire crosses is rst vertical downgoing

input wire insert a two input two output planar sub circuit that discards the value entering

from ab ove and passes the value entering from the left b oth to the right and down The

input to i from k is treated similarly with the obvious changes of orientation if kiAt

all other wire crossings insert a copy of the planar crossover circuit Note that given i and

j the wiring of cell i j is easily determined based on whether i j i j i is an input

to j etc Hence the reduction can b e p erformed in NC even if the original circuit is not

top ologically sorted

Remarks It is easy to see that monotone planar crossover networks do not exist so the

reduction ab ove cannot b e done in the monotone case In fact the monotone version of

PCVP is in LOGCFL NC DC DC Gol when all inputs app ear on one face of

the planar emb edding The more general problem where inputs may app ear anywhere is

also known to b e in NC Kos DK

A Arithmetic Circuit Value Problem ArithCVP

Given An enco ding of an arithmetic circuit with dyadic op erations and inputs

x x from a ring

n

Problem Do es on input x x output

n

Reference Ven

Hint Reduce NANDCVP to ArithCVP as follows true false and u v

u v where denotes the additive identity and denotes the multiplicative identityof

the ring

Remarks The problem is not necessarily in FP for innite rings like Z or Q since

intermediate values need not b e of p olynomial length It will b e in FP for any nite ring

and remains P hard in any ring It is also P complete to decide whether all gates in an

A Comp endium of Problems Complete for P Preliminary RCS Revision

k

n

arithmetic circuit over Z have smallv alues sayvalues of a magnitude for some

O

log

constant k Is in NC for circuits of degree where the degree ofanodeisfor

inputs and d d maxd d when the no de is pro duct resp ectively sum of the values

computed bytwo no des of degree d and d VSBR MRK

A MinPlus Circuit Value Problem MinPlusCVP

Given An enco ding of a min circuit and rational inputs x x

n

Problem Do es on input x x output a nonzero value

n

Reference Ven

Hint Reduce MCVP to MinPlusCVP as follows true false u v min u v

and u v min u v

Remarks The ab ove reduction works in any ordered semigroup with additive identity

and an elementsuchthat If there is a nonzero elementsuch that such

that eg in Z then reduce NANDCVP via u v minu v In a

wel lordered semigroup where is the minimum element one or the other of these cases

holds If the semigroup is innite the problem maynotbein P

A Inverting an NC Permutation InvNCPerm

n

Given An ninput noutput NC circuit computing a bijective function f f g

n n

f g and y f g

Problem Is the last bit of f y equal to

Reference Has Section Has

Hint

Remarks Not known to b e in P in general although the family of p ermutations used to

show P hardness is p olynomial time invertible

A Graph Theory

A Lexicographically First Maximal Indep endent Set LFMIS

Given An undirected graph G with an ordering on the vertices and a designated vertex v

Problem Is vertex v in the lexicographically rst maximal indep endent set of G

Reference Co o

Hint A pro of is given in Section

Remarks This is an instance of Problem A LFMIS is P complete for bipartite or

planar graphs restricted to degree at most Miy Karp observed that the completeness

th

of LFMIS implies that determining the i no de chosen byany deterministic sequential

algorithm for either LFMIS or LFMC Problem A is also complete Kar Computing

or approximating the size of the lexicographically rst maximal indep endent set is also P

complete see Section Karp and Wigderson gave the rst NC algorithm for nding a

maximal indep endent set KW subsequently improved by Luby Lub by Alon Babai

and Itai ABI and by Goldb erg and Sp encer GS These algorithms do not compute

the lexicographical ly rst maximal indep endentset

Part I I P Complete Problems

A Lexicographically First Maximal Clique LFMC

Given An undirected graph G with an ordering on the vertices and a designated vertex v

Problem Is vertex v in the lexicographically rst maximal clique of G

Reference Co o

Hint Finding a maximal clique is equivalent to nding a maximal indep endent set in the

complement graph of G Problem A

A Alternating Graph Accessibility Problem AGAP

Given A directed graph G V E a partition V A B of the vertices and designated

vertices s and t

Problem Is apath s t true where apath is dened as follows Vertices in A are uni

versalthose in B are existential Such a graph is called an alternating graph or an

ANDnOR graph Then apath x y holds if and only if

x y or

x is existential and there is a z with x z E and apath z y or

x is universal and for all z with x z E apath z y holds

Reference CKS Imm Imm

Hint Reduce AMCVP Problem A to AGAP Create two existential no des and

Put edge x into E if input x is and edge x into E if input x is and gates are

i i i i

universal no des and or gates are existential no des Inputs to a gate corresp ond to children

in the alternating graph For output z apathz holds if and only if the output z is

Remarks The original pro of simulated an alternating Turing machine ATM directly to

showthatAGAP was complete for ATM log space Imm Since ASPACElog n P

CKS this showed AGAP was P complete to o When this problem is generalized to hi

erarchical graphs it remains in P provided the graph is breadthrst orderedsee L W

The pro of sketched ab ove also shows that the problem remains P complete when the parti

tion A B induces a bipartite graph When restricted to only existential no des the prob

lem is equivalent to the directed graph accessibility problemv ariously called GAPand

STCONand kno wn to b e complete for NL Sav Peterson p ersonal communication

x shows that the undirected version of AGAPisalso P complete When restricted

to undirected graphs with only existential no des this problem is equivalent to the undi

rected graph accessibility problemcalled UGAPor USTCONwhic h is known to b e

complete for the sp ecial case of nondeterministic log space known as symmetric log space

SL LP

A Hierarchical Graph Accessibility Problem HGAP

Given A hierarchical graph G V E and two designated vertices s and t A hierarchical

graph G G consists of k sub cells G i k Each sub cell is a graph that

k i

A Comp endium of Problems Complete for P Preliminary RCS Revision

contains three typ es of vertices called pins inner verticesand nonterminals The pins are

the vertices through which the sub cell can b e connected to from the outside The inner

vertices cannot b e connected to from the outside The nonterminals stand for previously

dened sub cells A nonterminal inside G has a name and a type The name is a unique

i

numb er or string The typeisanumber from i A nonterminal v of typ e j stands

for a copy of sub cell G The neighbors of v are in a onetoone corresp ondence with the

j

pins of G via a mapping that is sp ecied as part of

j

Problem Is there a path b etween s and t in the expansion graph of G An expansion

graph is a hierarchical graph expanded The graph is expanded by expanding cell G recur

k

sivelyTo expand sub cell G expand its sub cells G G recursively and replace each

i i

nonterminal of v of typ e j with a copy of the expansion of sub cell G

j

Reference LW

Hint The reduction is from the alternating graph accessibility problem Problem A

Remarks Hierarchical versions of the following problems are also P complete graph

accessibility in undirected graphs determining whether a directed graph contains a cycle

and determining whether a given graph is bipartite LW There are several other very

restricted versions of hierarchical graph problems that are P complete LW

A Restricted Chromatic Alternating Graph Accessibility Problem RCA

GAP

Given An alternating graph G V E two natural numb ers k and m where k m

log jV j a coloring c E fmg and twovertices s and t Note that the coloring is

an unrestricted assignment of colors to the edges It may assign the same color to several

edges incident to a common vertex See Problem A for the denition of an alternating

graph

Problem Are there k dierent colors i i fmg such that apath s t holds in

k

the subgraph of G induced by the edges with colors i i See Problem A for the

k

denition of apath

Reference LW

Hint There is a trivial reduction from the alternating graph accessibility problem Problem

A Memb ership of RCAGAP in P follows from memb ership of AGAP in P since there

k

are at most jV j p ossible sets of colors to try

Remarks The problem remains P complete if the vertices are restricted to b eing breadth

rst ordered LW When generalized to hierarchical graphs the problem b ecomes NP

complete LW

A Lexicographically First Vertex Coloring LFDVC

Given AgraphG V E with equal to the maximum degree of anyvertex in V and

an ordering v v on the vertices of V a designated color candavertex v

n

Problem Is vertex v colored with color c in the lexicographically least coloring of the

vertices of G The coloring uses at most colors If c is the color of v where

i i

c f g then each coloring corresp onds to a ary numb er and the least

i

coloring is welldened

Part I I P Complete Problems

Reference Lub

Hint Computing the lexicographically least coloring is easily done in p olynomial time by

examining eachvertex in order and coloring it with the smallest available color To show

completeness reduce NANDCVP to LFDVC The coloring will corresp ond to evaluating

the circuit in top ological order Let v v b e the gates of a circuit and assume without

n

loss of generality that the gates are numb ered in top ological order Each gate in the circuit

will b e represented byfourvertices in G with the order on the vertices induced from the

order of the gates Consider a nand gate v v v Intro duce three new vertices

i j k

v v v where v v app ear after v v and v app ears after all these but b efore v in the

j k i

i j j i

k k

v v top ological ordering The gate is then represented bytheedgesv v v v

j k

j i j

k

v One nal x is necessaryTokeep the degree down to a fanout tree and v v v

i

i i

k

may b e required on the output of each gate The resulting graph can b e colored with only

colors in the order fT F Xg even though might b e necessary for a dierent ordering

Remarks The problem of coloring is NP complete GJ For graphs that are not

an o dd cycle or complete a coloring can b e found in p olynomial time Bro oks theorem

see BM However this is not necessarily the lexicographically rst The vertex

coloring is NC reducible to nding a maximal indep endent set Lub but the maximal

indep endent set algorithm KW do es not pro duce the lexicographically rst maximal

indep endent set It is p ossible to color a graph with colors in NC KN HS BK

although the coloring pro duced is not the lexicographically rst There is an NC algorithm

for coloring planar graphs Nao

A High Degree Subgraph HDS

Given A graph G V E and an integer k

Problem Do es G contain a vertex induced subgraph with minimum degree at least k

Reference AMa

Hint Reduce AMCVP to HDS The pro of illustrated here is for k although it can

b e generalized to any xed k A true input k connected to gate i is represented

by a gadget with vevertices k v v v and k The edges in the gadget are v k

v k v v v k v v v k v k and k is connected to a vertex in the

gadget for gate i as describ ed b elow A false input is represented by a single vertex An

and gate i with inputs l and l and outputs l and l is represented by a fourteen vertex

gadget The gadget is comp osed of two of the gadgets used to represent true inputs and

an additional four vertices l and l lab el the vertices corresp onding to k in the true

input gadget w and w lab el the p ositions corresp onding to k in their resp ectivecopy

of the true input gadget The four additional vertices are lab eled l l w and w l l

and w are connected into a three clique w is connected to w w and w Inputs to gate

i are connected to l and l and the outputs l and l are connected to the appropriate

input p ositions of other gates The representation of an or gate is very similar to the and

gadget omitting w and connecting l l directly to w Finally there is a binary tree that

has as its leaves the vertices corresp onding to the k s of the true inputs and has the vertex

corresp onding to the output vertex of the circuit as its ro ot The computation of a HDS of

degree pro ceeds on this new graph so that HDS is nonempty if and only if the output of

the circuit is true

A Comp endium of Problems Complete for P Preliminary RCS Revision

Remarks Although not stated as a lexicographically rstproblem the HDS in a graph

is unique hence this is another instance of Problem A There is an NC algorithm

computing a HDS for k Let K G denote the largest k such that there is an induced

subgraph of G with minimum degree k For xed c consider nding an approxima

tion d such that K G d cK G For any c there is an NC algorithm for nding

d and for any c the problem of nding d is P complete AMa A complemen

tarylo w degree subgraph problem has also b een studied and for several natural decision

problems it is NP complete Gre Decision problems based on ordered vertex removal

relating to subgraph computations are also P complete Gre A sp ecial case of HDS is

the Color Index Problem VS given an undirected graph G V E is the color index

of G less than or equal to four The color index is the maximum over all subgraphs H

of Goftheminimum degree of H Asking if the color index is is the complementto

asking if there are any high degree subgraphs of order The original reduction is from

the ordered lowdegreevertex removal problem Problem A See remarks for Problem

A

A High Connectivity Subgraph HCS

Given A graph G V E and an integer k

Problem Do es G contain a vertex induced subgraph of vertex edge connectivityat

least k

Reference KSS Ser

Hint The reduction is from MCVP and is similar to that used to prove Problem A

the high degree subgraph problem is P complete

Remarks Approximation algorithms for this problem exhibit a threshold typ e b ehavior

Below a certain value on the absolute p erformance ratio the problem remains P complete for

xed k and ab ove that ratio there are NC algorithms to approximate the problem SS

Sp ecicallyleto b e the maximum size of a k vertexconnected induced subgraph of G

Then for c it is p ossible to nd in NC a vertex induced subgraph of size co

but for c this not p ossible unless NC P For edgeconnectivity the threshold

is c

A Ordered High Degree Vertex Removal OHDVR

Given An undirected graph G V Ewithanumb ering on the vertices in V and two

designated vertices u and v

Problem Is there an elimination order on V v v satisfying the prop erties that u

n

is eliminated b efore v and for i n v is the lowest numb ered vertex of maximum

i

degree in the i st remaining subgraph of G An elimination order is a sequence of

vertices ordered as they and their corresp onding edges are to b e deleted from the graph

Reference Gre

Hint The reduction is from NAND CVP with fanin and fanout restrictions to The

circuit is transformed directly into a graph The vertices in the graph are ordered so that

gates evaluating to false in the circuit are deleted rst in the instance of OHDVR A

sp ecial vertex of degree four is added and its removal order is compared with that of the

Part I I P Complete Problems

vertex corresp onding to the output gate of the circuit

A Ordered Low Degree Vertex Removal OLDVR

Given An undirected graph G V E with a numb ering on the vertices in V and two

designated vertices u and v

Problem Is there an elimination order on V v v satisfying the prop erties that u is

n

eliminated b efore v and for i n v is the lowest numb ered vertex of minimum degree

i

in the i st remaining subgraph of G

Reference VSGre

Hint This is the complementary problem to Problem A The problem dened in VS

is more restricted than the one presented here Their graphs are also required to havethe

prop ertythatu app ears b efore v in some minimum elimination sequence if and only if u

app ears b efore v in all minimum degree elimination sequences

A Ordered Vertices Remaining OVR

Given An undirected graph G V E with a numb ering on the vertices in V adesig

nated vertex u and an integer k

Problem Is there an elimination order on V v v satisfying the prop erties that

n

u v for some jn k and for i n v is the lowest numbered vertex of maxi

j i

mum degree in the i st remaining subgraph of G

Reference Gre

Hint The reduction is from Problem A

Remarks The ordered low degree subgraph memb ership problem is also P

complete Gre The problem here is to determine whether a designated vertex is in

a remaining subgraph when all vertices in that remaining subgraph have small degree

A Nearest Neighbor Traveling Salesman Heuristic NNTSH

Given A distance matrix D with entries d and two distinguished vertices s and l

ij

Problem Do es the nearest neighbor tour starting at s visit l as the last vertex b efore com

pleting the tour at s The nearest neighb or tour is a greedy heuristic that always cho oses

the nearest unvisited vertex as the next vertex on the tour

Reference KLS

Hint Reduce NNTSH to NAND CVP Without loss of generality assume the gates are

numb ered in top ological order Gate k with inputs i and i and outputs o and o is

replaced by the gadget describ ed b elow The gadget has vertices A i i i i o o o o

and B Let the triple x y z mean the distance b etween x and y is d The triples

k o i i i in the gadget are A B k A i k A i k i i

i o k o o o o k o o and o Bk Vertex B of gate k is con

nected to vertex A of gate k The distances b etween vertices that have b een left un

sp ecied are assumed to b e very large The edges b etween ivertices and those b etween

overtices represent inputs and outputs resp ectively An edge included not included in

the tour represents a true falsevalue true circuit inputs are chainedtogether and

A Comp endium of Problems Complete for P Preliminary RCS Revision

the tour b egins at the rst true input By inserting a new no de C b efore vertex B in

the last gadget and connecting B and C to the rst true input in the input chain the

tour constructed by the NNTSH is such that B C is visited last if and only if the circuit

evaluates to true false

Remarks The nearest merger nearest insertion cheap est insertion and farthest inser

tion heuristics are all P complete KLS The double minimum spanning tree and nearest

addition heuristics are in NC KLS

A Lexicographically First Maximal Subgraph for LFMS

Given A graph G V E with an ordering on V a designated vertex v and a polyno

mial time testable nontrivial hereditary prop erty A prop ertyisnontrivial if there are

innitely many graphs that satisfy the prop erty and at least one graph that do esnt A

prop erty is hereditary on induced subgraphs if whenever G satises so do all vertex

induced subgraphs

Problem Is v in the lexicographically rst maximal subgraph H of G that satises

Reference Miy

Hint Given a prop erty that is nontrivial and hereditary Ramseys theorem implies that

either is satised by all cliques or by all indep endentsetsofvertices This observation

combined with the facts that the lexicographically rst maximal clique problem Problem

A and the lexicographically rst maximal indep endent set problem are P complete are

used to show LFMS is P complete

Remarks The following are examples of prop erties that meet the criteria stated in the

problem bipartite chordal clique comparability graph edge graph forest indep endent

set outerplanar and planar Not all problems computing a lexicographically rst solution

are P complete For example the lexicographically rst top ological order problem is com

plete for NLOG Sho and the lexicographic low degree subgraph memb ership problem is

NP complete Gre

A Minimum FeedbackVertex Set MFVS

Given A directed graph G V E that is cyclical ly reducible dened b elow and a

designated vertex v

Problem Is v contained in the minimum feedback set of G that is computed by the

algorithm given in WLS

Reference BDAP

Hint We review some terminology WLS A no de z of G is dead locked ifthereisa

directed path in G from z toanode y that lies on a directed cycle The associatedgraph

of node x with respect to G AGx consists of no de x and all no des of G that are not

deadlo cked if x is removed from G A directed graph is cyclical ly reducible if and only if

there exists a sequence of no des y y such that each of the graphs AG y is

k i i

cyclic where G G and G G AG y for i k

i i i i

A set is called a feedback vertex set if it contains at least one vertex from every cy

cle Kar It is minimum if no other feedbackvertex set has fewer elements Wang

Lloyd and Soa gave a p olynomial time algorithm for computing feedback sets in cyclically

Part I I P Complete Problems

reducible graphs WLS Thus MFVSP is in P The reduction is from MCVP Let

g g denote an instance of MCVP including inputs where g is the output gate

k k

From a graph G is constructed as follows

with each g asso ciate no des g and g

i i

i

for each input g if g is true false add a lo op edge to g g

i i

i

i

for each and gate g with inputs i and j add edges g i i j j g g i

i g g j and j g to G

for each or gate g with inputs i and j add edges g i i j j g g i i g

g j and j g toG

add edges g g and g g

k k k k

It is easy to see that G is cyclically reducible Set v g to complete the reduction

k

Remarks The question of whether a graph has a feedbackvertex set of size k is NP

complete Kar Bovet De Agostino and Petreshci give an algorithm for nding a mini

mum feedback set that requires O k log n time and O n pro cessors on a CREWPRAM

where k denotes the size of the minimum feedback set Greenlawproved that a related prob

lem the lexicographically rst maximal acyclic subgraph problem is P complete Gre

Ramachandran proved that nding a minimum weight feedbackvertex set in a reducible

ow graph with arbitrary weights is P complete Ram She also proved the following

four problems are NC equivalent nding a minimum feedback arc set in an unweighted

reducible ow graph nding a minimum weight feedback arc set in a reducible ow graph

with unary weights on the arcs nding a minimum weight feedbackvertex set in a reducible

ow graph with unary weights on the vertices and nding a minimum cut in a ow network

with unary capacities Ram

A Edge Maximal Acyclic Subgraph EMAS

Given A directed graph G V E with an ordering on the edges and a designated

edge e

Problem Is e contained in the edge maximal acyclic subgraph

Reference Gre

Hint The edge maximal acyclic subgraph is dened to b e the subgraph computed byan

algorithm that builds up the subgraph by pro cessing edges in order It adds an edge to the

subgraph if its inclusion do es not intro duce a cycle The reduction is from NOR CVP A

gadget is designed that replaces each gate in the circuit A gadget for gate g with inputs i

i and outputs o o has three no des g top g mid and g b ot The edges in the gadget

are g topi mid g topi mid g midi b ot g midi b ot g b oto top

g topg mid g midg b ot g b oto top and g b otg top The rst ve edges

are upwarding p ointing edges and are ordered rst The last four edges are ordered within

the gadget as listed true input i to gate g is represented by a no de with an edge to g top

false input i to gate g is represented bytwo no des forming a cycle with g top The edges

A Comp endium of Problems Complete for P Preliminary RCS Revision

are ordered so that the edge leading into g top is not put in the set the algorithm con

structs The edges of the output gate are groundedOne of the edges lea ving g b ot

out

is used as the designated edge e

Remarks The decision problem Is the edge maximal subgraph of size k is also P

complete Gre The feedback arc set problem Karisanequivalentformulation of the

maximum acyclic subgraph problem Approximation algorithms for this problem are im

p ortant b ecause there are very few classes of graphs for which the problem is known to b e in

P BS Greenlawproves decision problems based on several other natural approximation

algorithms are P complete Gre He also gives two approximation algorithms that are

in NC Berger has an NC approximation algorithm that assuming the input graph do es

not contain twocycles generates a subgraph containing more than half the arcs Ber

Ramachandran proved that nding a minimum weight feedback arc set in a reducible ow

graph with arbitrary weights on the arcs in P complete Ram

A General Graph Closure GGC

Given An undirected graph G V E a subset E V V with E E and a

designated edge e u v E

Problem Is e in the general closure G G E ofG That is the graph obtained from G

by rep eatedly joining nonadjacent pairs of vertices u and v whose degree sum is at least

jV j and suchthatu v E The edges in E are called admissible edges

Reference Khu

Hint An O n algorithm solving the problem is given in Khu The reduction is

from MCVP Problem A The idea is to replace gates in the circuit by gadgets We

will describ e only the gadgets for inputs and and gates The gadgets for or gates are

similar A true falsevalue is asso ciated with the output vertex of a gadget if its degree

has increased by the addition of an admissible edge remained the same The gadget is

constructed so that valuesdo not propagate bac k up through the circuit N will denote

the total numberofvertices contained in all the gadgets An additional N vertices are

added and connected so as to double the degrees of the vertices in the construction of the

graph containing gadgets The total degree of the graph constructed is N Atrue input

is represented bytwovertices with an admissible edge b etween them and the degree of

eachvertex is N A false input is represented similarly except on the output side the

vertex has degree N The gadget representing an and gate consists of thirteen

k

vertices We describ e the upp er left part which consists of vevertices rst Well call

the vertices and Vertex has a connection from one of s inputs Vertices

k

and havedegree N and vertices and have degree N The admissible edges

are and The upp er right part of the gadget is similar

with vertices playing the roles of Vertices and are connected to vertex

via admissible edges Vertex has degree N Vertex is connected to the outputs

of the gadget These are vertices and They b oth are of degree N The gadgets are

connected in the obvious manner The circuit evaluates to true if and only if the admissible

edge of the gadget corresp onding to the output gate is added to G

Remarks The complexity of the general graph closure problem in which E V V E is op en see Problem B

Part I I P Complete Problems

A Search Problems

A Lexicographically First Maximal Path LFMP

Given AgraphG V E with a numb ering on the vertices and two designated vertices

s and t

Problem Is vertex t on the lexicographical ly rst maximal path in G b eginning at sA

maximal path is a path that cannot b e extended b ecause any attempt at extending it will

result in an encounter with a no de that is already on the path The lexicographically rst

maximal path is the maximal path that would app ear rst in the alphab eticallisting of

all paths from s where the alphab etizing is done with resp ect to the vertex numb ers

Reference AMb

Hint The reduction is from a version of CVP consisting of not and or gates The key

idea is the construction of a subgraph called a latchAlatch consists of six no des connected

in a rectangular fashion The latches are ho oked together and lab eled in a clever manner

A latch that has b een traversed not traversed in the construction of the lexicographically

rst maximal path indicates a true falsevalue for the corresp onding gate Vertex t is

a sp ecial vertex in a latch corresp onding to the output gate

Remarks LFMP remains P complete when restricted to planar graphs with maximum

degree three If the maximum degree of anyvertex in G is at most then there is an

algorithm that can nd a maximal path in O log n time using n pro cessors AMb

There is also an NC algorithm for nding a maximal path in planar graphs AMb The

complexity of the general problem of nding a maximal path is op en AMb although

known to b e in RNC And

A Lexicographically First Depth First Search Ordering LFDFS

Given AgraphG V E with xed ordered adjacency lists and two designated vertices

u and v

Problem Is vertex u visited b efore vertex v in the depth rst searchofG induced bythe

order of the adjacency lists

Reference Rei

Hint Follows easily from Problem A since the leftmost path in the lexicographically

rst depth rst search tree is the lexicographically rst maximal path And Reif

Rei gives a direct reduction from NOR CVP to DFS taking advantage of the xed

order by which the adjacency lists are examined We present the directed case from

which the undirected case is easily derived Without loss of generality assume gates

are numb ered in top ological order The gadget describ ed b elow replaces nor gate i

having inputs i and i and outputs to gates j and j The gadget has vertices

enter iini i ini i si outi outi ti and exiti Let the triple x y z

th

denote a directed edge x y with y app earing z on xs adjacency list The gadget

has triples enter iini i ini i ini i ini i si siouti

outi si outi inj i outi outi

outi inj i ti outi enter iti tiexiti si exiti

and exiti enter i Additionallyinj iouti and inj iti are

triples connected to the gadget true inputs are chainedtogether The lexicographic

A Comp endium of Problems Complete for P Preliminary RCS Revision

DFS of the graph constructed visits vertex sn where n corresp onds to the output gate

b efore after tn if and only if the circuit evaluates to true false

Remarks The directed case can b e easily reduced to the undirected case The reduc

tion is dep endent on the adjacency lists xing the order in which the adjacent no des

are examined The problem remains op en if this constraint is relaxed For example

the problem remains op en for graphs presented with all adjacency lists sorted in order

of increasing vertex number The problem remains P complete if the input is sp ecied

by a xed vertex numb ering And Greb Anderson showed that computing just the

rst branch of the lexicographically rst DFS tree called the lexicographically rst max

imal path is P complete And see Problem A Computing the LFDFS tree in

planar graphs is P complete as well And In RNC it is p ossible to nd some depth

rst vertex numb ering and the depthrst spanning tree corresp onding to it see Problem

B AA AAK Computing a depthrst vertex numb ering for planar graphs is in

NC Smi HY Computing the lexicographically rst depthrst numb ering for DAGs

is in NC Grear dlTK dlTK Determining whether a directed spanning tree of a

general graph has a valid DFS numb ering is in NC SV

A BreadthDepth Search BDS

Given A graph G V Ewitha numb ering on the vertices and two designated vertices

u and v

Problem Is vertex u visited b efore vertex v in the breadthdepth rst search HSofG

induced by the vertex numb ering A breadthdepth rst search starts at a no de s and visits

all children of s pushing them on a stack as the search pro ceeds After all of ss children

have b een visited the searchcontinues with the no de on the top of the stack playing the

role of s

Reference Greb Grear

Hint The pro of sketched b elow is due to Anderson And Reduce LFDFS to BDS

Insert a new vertex b etween every pair of connected no des in the original graph Supp ose

in the original graph corresp onding to the circuit that vertex u has children v v and

i k

that these vertices are visited in this order by the greedy DFS algorithm Let c c be

k

the no des inserted by the reduction b etween u and v u and v resp ectively The

i i k

c s are assigned numb ers so that in increasing order they are listed as c c The vertex

i k

numb ers assigned to these no des are sp ecied to reversethe breadthdepth searc h in the

new levels In this way the DFS order of the original graph can b e maintained Vertex u is

visited b efore vertex v in an instance of LFDFS if and only if the vertex corresp onding to

u is visited b efore the vertex corresp onding to v in the constructed instance of BDS

A Stack Breadth First Search SBFS

Given A graph G V Ewitha numb ering on the vertices and two designated vertices

u and v

Problem Is vertex u visited b efore vertex v in the stack breadth rst search of G induced

by the vertex numb ering A stack breadth rst search is a breadth rst searchthatis

implemented on a stack The no des most recently visited on a new level are searched from

Part I I P Complete Problems

rst at the next level

Reference Greb Grear

Hint The reduction is from SAMCVP to SBFS Sort the inputs no des and assign false

inputs lower numb ers than true inputs In increasing order let f f t t denote

k m

the ordering induced bythisnumb ering A new vertex e is intro duced and given a number

between f and t For eachgateavertex is intro duced and its connections in the circuit

k

are maintained in the graph b eing constructed A start vertex s is added and connected to

all inputs Additionally a new chain of vertices starting from s is added The vertices in

this chain are s e e e where D denotes the depth of the circuit The searchorder

D

sp ecied by stack breadth rst search is suchthatvertex e for l odd even corresp onding

l

to an or andlevel in the instance of SAMCVP is visited b efore vertex v if and only if

the gate corresp onding to vertex v evaluates to false true in the circuit

Remarks The lexicographic breadth rst search problem which has a natural implemen

tation on a queue is dened as follows given a graph G with xed ordered adjacency lists

is vertex u visited b efore vertex v in the breadth rst searchofG induced by the order of

the adjacency lists This problem is in NC Greb dlTK dlTK

A Alternating Breadth First Search ABFS

Given A graph G V E with E partitioned into twosets M and U a designated vertex

v and a designated start vertex s

Problem Do es vertex v get visited along an edge from the set M during an alternating

breadth rst search of G An alternating breadth rst search which has applications in

some matching algorithms is a breadth rst search in which only edges in the set U M

can b e followed in going from even o dd to o dd even levels

Reference And And

Hint Andersons pro of was from a version of CVP comp osed of or and not gates The

reduction we presentisfromNANDCVP Let t t denote true inputs f f

k l

denote false inputs and g g denote nand gates A new vertex s is created from

m

where the search will originate For each t i k two new vertices t and t are

i

i i

intro duced for each f i l a new vertex f is intro duced and for each g i m

i i

i

avertex v is intro duced f is connected to s by an edge in U and to v where j is such

i j

i

that f was input to gate g by an edge in M t is connected to s byanedgein U t is

j

i i i

connected to t by an edge in M andt is connected to v wherej is such that t was input

j

i i i

to gate g If gate g has outputs to gates g and g thenv v andv v are edges in

j i j h i j i h

M For each of these gates receiving inputs from g there are two additional vertices For

i

g they are called v and v v v is an edge in U v v isin M and v v isin

j ij i ij ij j

ij ij ij

U For gate g similar vertices and edges are added The circuit evaluates to true false

h

if and only if the vertex corresp onding to the output gate of the circuit is visited along an

edge in M U

Remarks The matching constructed by the search is not necessarily maximal The prob

lem of nding a maximum matching is in RNC MVV The problem of nding a p erfect

matching is also in RNC KUW MVV

A Comp endium of Problems Complete for P Preliminary RCS Revision

A Combinatorial Optimization and Flow Problems

A Linear Inequalities LI

Given An integer n d matrix A and an integer n vector b

Problem Is there a rational d vector x such that Ax b It is not required to

nd suchan x

Reference Co oVala Kha

Hint LI is in P byKha The following reduction of CVP to LI is due to Co o

If input x is true false it is represented by the equation x x

i i i

A not gate with input u and output w computing w u is represented by the

inequalities w u and w

An and gate with inputs u v computing w u v is represented by the inequalities

w w u w v andu v w

An or gate is represented by the inequalities w u w v w andw u v

Note for any gate if the inputs are or the output will b e or To determine the

output z of the circuit add the inequalities required to force z If the system has a

solution then the output is true and otherwise the output is false

A Linear Equalities LE

Given An integer n d matrix A and an integer n vector b

Problem Is there a rational d vector x such that Ax b

Reference Co oVala Kha

Hint LE is NC reducible to LI since Ax b and Ax b if and only if Ax bThus

LE is in P For completeness an instance of LI can b e reduced to LE as follows for each

inequality in LI there is a corresp onding equality in LE with an additional slackv ariable

that is used to make the inequalityinto an equality

Remarks If LE is restricted so the co ecients of A and b are either or then LE

is still P complete This follows from the reduction given by Itai Ita The restricted

version of LE is denoted LE

A Linear Programming LP

Given An integer n d matrix Aaninteger n vector b and an integer d vector c

Problem Find a rational d vector x such that Ax b and cx is maximized

Reference DLR Kha DRVala

Hint LP is not in P but is in FP by Kha Reduce LI to LP by picking anycostvector

csay c and checking whether the resulting linear program is feasible

Remarks The original reduction in DLR is from HORN Problem A to LP

In DR LP and LI are shown to b e log space equivalentby reducing LP to LI using

rational binary searchPap Rei to nd the value of the maximum and an x that

Part I I P Complete Problems

yields it However it is not clear how to p erform this reduction in NC Since LP and

LI are complete via NC reductions though there mustbeaNC reduction b etween the

two problems Although we know that LP and LI are NC equivalent the NC reduction

between them is not an obvious one It is also P hard to approximate cx to within any con

stant fraction even given a feasible solution x Anne Condon p ersonal communication

April reduction is from Problem A

A Maximum Flow MaxFlow

Given A directed graph G with each edge lab eled with a capacity c two distinguished

i

vertices source s and sink t andavalue f

Problem Is there a feasible owofvalue k ie is the value of the maximum owinto the

sink f

Reference GSSLW

Hint The rst P completeness pro of for a decision problem derived from maximum ow

was for the problem of determining whether the ow is o dd or even GSS We give this

reduction The pro of given in LW for the more natural threshold problem stated ab ove

is similar The reduction is from AMCVP Problem A to MaxFlow Gates v and

i

of G G has additional no des and edges e connections e of are asso ciated with no des v

ij

ij i

s t and an overow edge for each v Each edge e of has a capacity and a ow asso ciated

ij

i

i i

with it This capacityis and the owis if gate v is true and otherwise A no de v

i

i

j k

with inputs v and v has a maximum p ossible inowof and outow to other gates

j

k

i

of d d is the outdegree of v The remaining ow is absorb ed by the overow edge from

i

j k i

with capacity d This overow edge is directed toward t in case v is an and v

i

i

gate and toward s in case v is an or gate Thus the no des must b e top ologically ordered

i

with the output rst and the inputs last and the output gate must b e an or gate Note

that all edge capacities are even except the one from v to tThus the maximum owfor

G is o dd if and only if outputs true

Remarks This reduction pro duces exp onential edge capacities in G In a network with

edge capacities expressed in unary computing the magnitude of the maximum owisin

RNC Fea and a metho d for nding the owin RNC is also known KUW If the

network is restricted to b eing acyclic MaxFlow remains P complete Ram Flows in

planar networks can b e computed in NC JV Two commo dityow CF is dened

like MaxFlow except there are two sources and two sinks and there are two separate ows

functions for the commo dities Since MaxFlow is a sp ecial case of CF and CF is in P by

results of Itai Ita and Khachian Kha it follows that CF is P complete Itai dened

several other variants of CF that are also P complete They are dened b elow l uCF

is a case of CF in which there are lower and upp er b ounds on the capacity of each edge

Selectivel uCF is dened to b e a CF with lower and upp er b ounds on the sum of

the twoows on each edge Stein and Wein SWshow that althought there is an RNC

algorithm to approximate maximum ow approximating the minimum cost maximum ow

is P complete

A Comp endium of Problems Complete for P Preliminary RCS Revision

A Homologous Flow HF

Given A directed graph G with each edge v w lab eled with a lower and upp er b ound

on ow capacity l v wuv w andtwo distinguished vertices source s and sink t

Problem Is there a feasible ow in the network A feasible ow is one in which the ow

assigned to each arc falls within the lower and upp er b ounds for the arc A homologous

ow is a ow in which pairs of edges are required to have the same ow

Reference Ita

Hint It follows that HF is in P by the results of Itai Ita and Khachian Kha We

describ e the reduction given by Itai Ita and note that it is a log space reduction The

P

m

reduction is from LE Let a x b fori n b e an instance of

ij j i

j

i

LE For f g let J fj j a g Another formulation of the original

ij

P P

equations is x x b fori n There are n sections in the ow

i i

j j i

j J j J

i i i i i i i

network constructed and each one has m vertices fv v y z J J J gFor

m

i i i

if j J then add v J asanonrestrictededge one with lower b ound

j

i i

z with lower and upp er capacities equal and upp er b ound to the network Add J

i i i i i i i i

to b J y and J y are homologous nonrestricted edges J z and J y are

i

nonrestricted edges An additional no de z is added as a source and z is the sink For

n

n n

each j z v z v z v are pairwise nonrestricted homologous edges Let f

j j j

denote the ow given a solution x to the equations a feasible owis x f z v

j

j

n

n

f z v and given a feasible ow it is easy to construct solution x

j

A Lexicographically First Blo cking Flow LFBF

Given A directed acyclic graph G V E represented by xed ordered adjacency lists

with each edge lab eled with a capacity c and two distinguished vertices source s and

i

sink t

Problem Is the value of the lexicographically rst blocking ow o dd A blo cking owisa

ow in whichevery path from s to t has a saturatededge an edge whose ow is equal to

its capacity The lexicographically rst blo cking ow is the ow resulting from the standard

sequential depth rst search blo cking ow algorithm

Reference AMa

Hint The reduction given in Problem A can b e easily mo died to show this problem

is P complete

Remarks The problem of nding the lexicographically rst blo cking owina layered

network is also P complete A layered network is one in which all source to sink paths

have length Cheriyan and Maheshwari also giveanRNC algorithm for nding a blo cking

owinalayered network CM

A First Fit Decreasing Bin Packing FFDBP

Given A list of n items v v where each v is rational number between and and

n i

two distinguished indices i and b

th th

Problem Is the i item packed into the b bin by the rst t decreasing bin packing heuristic

Part I I P Complete Problems

Reference AMW

Hint Reduce AMCVP Problem A to FFDBP Without loss of generalitywe can

assume the gates are numb ered in top ological order The reduction transforms the

n

sequence into a list of items and bins Let in and n

n i

and T F denote any item of size We describ e how to construct the list of

i i i i

items and bins For and gate with outputs to gates and construct bins of size

i j k

and and items of size

i i i j i k i i i i i

and For an or gate with outputs to gates and construct bins of size

i i j k

and and the same items as for the and gate The output

i i i j i k

gate is treated sp ecially and has bins of size and and items of size

n n n n n n

and For gates receiving a constant circuit input a T F is removed if the gate

n i i

receives a false true input The lists of bins are concatenated in the order of their

corresp onding gate numb ers and similarly for the items To get unit size bins let u u

q

b e the nonincreasing list of item sizes and let b b b e the list of variable bin sizes as

r

constructed ab ove Let B max b and C r B For i r set v C iB b

i i i

if i r and C ib otherwise Packing these r items into r bins of size C has the aect of

th

leaving b space in the i bin By concatenating the uand v item lists and normalizing

i

the bin sizes a rst t decreasing bin packing of the items will place the item corresp onding

to the second T in s list into the last bin if and only if the circuit evaluates to true

n n

Remarks The problem remains P complete even if unary representations are used for

the numb ers involved This is one of the rst such problem where large numb ers do not

app ear to b e required for P completeness in contrast see MaxFlow Problem A The

problem of determining if I is the packing pro duced by the b est t decreasing algorithm is

also P complete AMW In AMW there is an NC algorithm that pro duces a packing

within of optimal This is the same p erformance as for rst t decreasing

A General List Scheduling GLS

Given An ordered list of n jobs fJ J gapositiveinteger execution time T J for

n i

each job and a nonpreemptiveschedule L The jobs are to b e scheduled on two identical

pro cessors

Problem Is the nal oset pro duced by the list scheduling algorithm nonzero The nal

oset is the dierence in the total execution time of the two pro cessors

Reference HM

Hint Reduce NOR CVP to GLS Without loss of generality assume the gates in the

instance of NOR CVP are numb ered in reverse top ological order The input wires to gate i

i i

are numbered and The output wire of gate the overall circuit output is lab eled

Let V b e the sum of the lab els on the output wires of gate iFor gate ijobsare

i

i i

intro duced with the following execution times job at jobs at and

i

jobs at V The initial job has execution time equal to the sum of the lab els of all

i

true input wires The remaining jobs are listed in descending order of gate numb er The

nal oset will b e if and only if the output gate i is true false

O

Remarks The problem is in NC if the job times are small that is n NC algorithms for

scheduling problems with either intree or outtree precedence constraints are known HM

HM

A Comp endium of Problems Complete for P Preliminary RCS Revision

A Lo cal Optimality

A MAXFLIP Verication MAXFLIPV

Given An enco ding of a Bo olean circuit constructed of and orandnot gates plus

inputs x x x The circuits m output values y y y

n m

Problem Is the circuits output a lo cal maximum among the neighbors of x when y is

viewed as a binary numb er The neighbors of x are vectors of length n whose Hamming

distance diers from x by That is they can b e obtained from x by ipping one bit

Reference JPY

Hint The problem is easily seen to b e in P The reduction is from the MCVP Problem

A Let denote an instance of MCVP with input x x Construct an instance of

n

MAXFLIP as follows The new circuit is the same as except for a mo dication to the

input Add a latchinput that is andd with eachofs inputs b efore they are fed into

later gates of Set the latch input to value The output of the circuit constructed will

b e The input x x will b e lo cally optimal if and only if the output is when the

n

latch input is This is true if and only if the output of on its input is

Remarks The complementary problem called the FLIP verication problem in which

the output is minimized is also P complete JPY The general problems MAXFLIP and

FLIP are PLS complete JPY PLS stands for p olynomially lo cal search

A Lo cal Optimality KernighanLin Verication LOKLV

Given A graph G V Ewithweights w e on the edges and a partition of V into two

equal size subsets A and B

Problem Is the cost of the partition cA B a lo cal optimum among the neighbors of the

partition The cost of the partition is dened to b e the sum of the costs of all edges going

between the sets A and B We follow the presentation in JPY to dene the neighb ors

A swap of partition A B is a partition C D suchthatC D is obtained from A B

byswapping one elementofA with an elementofB Theswap C Disa greedy swap if

cC D cA B is minimized over all swaps of A B If C D is the lexicographically

smallest over all greedy swaps then C Dissaidtobethelexicographical ly greedy swap

of A B A sequence of partitions A B each obtained bya swap from the preceding

i i

partition is monotonic if the dierences A A and B B are monotonically increasing

i i

A partition C D is a neighbor of A B if it o ccurs in the unique maximal monotonic

sequence of lexicographic greedy swaps starting with A B

Reference JPY

Hint The reduction is from MAXFLIPV Problem A

Remarks A problem called the weak local optimum for KernighanLin verication problem

in whichtheneighborhoods are larger is also P complete JPY The general versions of

these problems local optimality KernighanLin and weak local optimality KernighanLin

are b oth PLS complete JPY

A Unweighted NotAllEqual Clauses SATFLIP UNSATFLIP

Given A Bo olean formula F in CNF with literals p er clause and a truth assignment s

Part I I P Complete Problems

Eachclausehasaweight of The clauses are notal lequals clauses with p ositive literals

A truth assignment satisesa clause C under the notallequals criterion if it is such that

C has at least one true and one false literal

Problem Is the assignment s the maximum cost assignment of F over all neighbors of

s The cost of the assignment is the sum of the weights of the clauses it satises The

neighb ors of s are assignments that dier from s in one bit p osition

Reference PSY SY

Hint The reduction is from NOR CVP

Remarks The weighted version of the problem is PLS complete PSY

A Unweighted MAXCUTSWAP UMS

Given A graph G V Ewithweights of size on the edges and a subset S V

Problem Is S the maximum cost subset of no des where the cost is the sum of the weights

of the edges leaving no des in S over all neighbors of S A neighbor of S is a set of size jS j

whose symmetric dierence with S contains one no de

Reference PSY SY

Hint The reduction is from UNSATFLIP Problem A

Remarks The weighted version of the problem is PLS complete PSY

A Unweighted SATFLIP USATFLIP

Given A Bo olean formula F in CNF with literals p er clause and a truth assignment s

Each clause has a weightof

Problem Is the assignment s the maximum cost assignment of F over all neighbors of s

The cost of the assignment is the sum of the weights of the clauses it satises The neighb ors

of s are assignments that dier from s in one bit p osition

Reference PSY SY

Hint The reduction is from UMS Problem A

Remarks The weighted version of the problem is PLS complete PSY

A Unweighted SWAP USWAP

Given A graph G V Ewithn no des and a set S of n no des

Problem Is S the minimum cost set of n no des where the cost is the sum of the weights

of the edges leaving S over all neighbors of S A neighbor of S is a set of n no des whose

symmetric dierence with S consists of two no des

Reference PSY SY

Hint The reduction is from UMS Problem A

Remarks The weighted version of the problem is PLS complete PSY

A Unweighted STABLE NET USN

Given A graph G V E whose edges all haveweights of and an assignment B of

lab els x f g to eachnode i i

A Comp endium of Problems Complete for P Preliminary RCS Revision

P

Problem Do es the assignment B yield maximum cost x x over all i j over all

i j

ij

neighbors of B A neighbor of B is an assignment of lab els that diers from B on only one

no de

Reference PSYSY

Hint The reduction is from UMS Problem A

Remarks The weighted version of the problem is PLS complete PSY

A Logic

A Unit Resolution UNIT

Given A Bo olean formula F in conjunctive normal form

Problem Can the empty clause b e deduced from F by unit resolution A unit is a

clause with only one term For example the unit resolventofF A B B and

m

the unit G A is B B

m

Reference JL

Hint Jones and Laaser provide a p olytime algorithm for unit resolution JL To show B

follows from the assumption A A negate B add it to the set of clauses and derive the

m

empty clause Reduce CVP to UNIT as describ ed b elow A gate in the circuit v v v

k i j

is represented by the clauses of v v v that is v v v v v v v

k i j k i k j k i j

Similarly v v v is represented by the clauses of v v v and v v is

k i j k i j k i

represented by the clauses of v v

k i

Remarks Under the appropriate denitions it is known that approximating this problem

is also P complete SS

A Horn Unit Resolution HORN

Given A Horn formula F that is a conjunctive normal form CNF formula with each

clause a disjunction of literals having at most one p ositive literal p er clause

Problem Can the empty clause b e deduced from F by unit resolution

Reference DLR JL

Hint Reduce an arbitrary Turing machine to a CNF formula as in Co ob All of the

f abc

a b c

clauses are Horn clauses Most clauses are of the form P P P P

it it it

it

a

where P t is true if at time t tap e cell i contains symbol a or symbol a s if the tap e

i

head is over the cell and the Turing machine is in state s The function f a b c dep ends

on the Turing machine For cells i ii containing symbols a b c the value of cell i

at the next time step is f a b c Alternatively reduce CVP to HORN as for UNIT The

clauses for v v v and for v v are already in Horn form For v v v the

k i j k i k i j

clauses of v v v are not in Horn form but replacing the or byanand gate and

k i j

asso ciated not gates using DeMorgans laws results in a set of Horn clauses

A Prop ositional Horn Clause Satisability PHCS

Given AsetS of Horn clauses in the prop ositional calculus

Problem Is S satisable

Part I I P Complete Problems

Reference Pla Kas

Hint The reduction is straightforward from the alternating graph accessibility problem

Problem A

Remarks The problem remains P complete when there are at most literals p er

clause Pla Plaisted has shown that two problems involving pro ofs of restricted depth

are also P complete They are the two literal Horn clause unique matching problem and

the three literal Horn clause problem Pla

A Relaxed Consistent Lab eling RCL

Given A relaxed consistent lab eling problem G consisting of a set of variables V

fv v g and a set of lab els L fL L gwhere L consists of the p ossible lab els

n n i

for v A binary predicate P where P x y if and only if the assignmentoflabel x

i ij

to v is compatible with the assignment of lab el y to v A designated variable P and a

i j

designated lab el f

Problem Is there a valid assignment of the lab el f to P in G

Reference Kas GHR

Hint The original reduction is from the prop ositional Horn clause satisability problem

Problem A Kas The reduction wesketch is from NAND CVP A variable is intro

duced for each circuit input The variable must have lab el if the circuit input is true

false We view each nand gate as b eing represented by three variables Consider nand

gate k with inputs i and j and outputs s and t The variables for k will b e denoted k L

k

and R The p ossible lab els for a nand gate variable k are T T andF T and

k

T are used to denote true values and F is used to denote a false value The p ossible

lab els for variables L and R are and The constraints for variable k with its inputs

k k

are as follows P P T P T P F P

ik ik ik ik jk

P T P T and P F The constraints on L and R are the

jk jk jk k k

same When k has anylabel l from fTT g then P l When k has a lab el l from

kL

k

fFg then P l All other p ossible lab elings are not allowed The constraints

kL

k

involving nand gate s t use L R Notice since inputs have only one p ossible lab el

k k

they must b e assigned this lab el The remainder of the lab eling is done so that the circuit

gets evaluated in a top ological manner Let o denoted the numb er of the output gate of the

nand circuit instance There is a valid assignment of lab el to no de L if and only if the

o

output of gate o is true

Remarks The general consistent lab eling problem is NP complete

A Generability GEN

Given A nite set W a binary op eration on W presented as a table a subset V W

and w W

Problem Is w contained in the smallest subset of W that contains V and is closed under

the op eration

Reference JL

Hint The reduction is from unit resolution Problem A Dene a b to b e the unit

resolution of clauses a and bLetW b e all the sub clauses of a formula F in an instance of

A Comp endium of Problems Complete for P Preliminary RCS Revision

UNIT and V b e all of its clauses Let w b e the empty clause

Remarks If is asso ciative GEN is complete for NSPACElog n The problem remains

in P even with more than one op eration Under the appropriate denitions it is known

that approximating this problem is also P complete SS

A Path Systems PATH

Given A path system P X R S T where S X T X and R X X X

Problem Is there an admissible no de in S A no de x is admissible if and only if x T

or there exists admissible y z X suchthatx y z R

Reference Co o JL

Hint Reduce GEN to PATH by dening x y z R if and only if x y z

Remarks This is the rst problem shown to b e log space complete for P The original

pro of byCookdoesadirectsimulation of a Turing machine Co o Under the appropriate

denitions it is known that approximating this problem is also P complete SS

A Unication UNIF

Given Two symb olic terms s and t Each term is comp osed of variables and function

symb ols A substitution for x in a term u is the replacement of all o ccurrences of a variable

x in u by another term v

Problem Is there a series of substitutions that unify s and t That is gives s t

The two terms are called uniable if sucha exists

Reference DKMYas DKS

Hint The reduction given in DKM is from MCVP The reductions given in DKS

are from NAND CVP

Remarks Unrestricted unication is also P complete DKM Unrestricted unication

is where we allow substitutions to map variables to innite terms It is convenientto

represent terms as lab eled directed acyclic graphs A term is linear if no variable app ears

more than once in the term The following two restricted versions of unication are also

b oth P complete a b oth terms are linear are represented by trees and have all function

symb ols with arity less than or equal to two b b oth terms are represented by trees no

variable app ears in b oth terms eachvariable app ears at most twice in some term and all

function symb ols have arity less than or equal to two DKS A restricted problem called

term matching can b e solved on a CREWPRAM in randomized time O log n using M n

pro cessors where M n denotes the complexityofann n matrix multiplication DKS

Aterms matchesatermt if there exists a substitution with s t Vitter and Simons

p

n time parallel algorithms for unication and some other P complete problems VS give

A Logical Query Program LQP

Given An extended logic program P and a ground clause C of the form

pxq y q y A ground clause is one in which all arguments egx andy

k k i

ab ove are constants An extendedlogic program is a basic logic program plus an extensional

data base instance A basic logic program is a nite set of rules A rule is a disjunction

Part I I P Complete Problems

of literals having exactly one p ositive literal called the head The negative literals in the

clause are called subgoals The set of predicate symb ols that app ear only in subgoals is

called the Extensional DatabaseorEDBAn EDB fact is an EDB predicate with constants

as arguments An EDB instance is a nite set of EDB facts C is a theorem of P if the

head of C is derivable from its subgoals in P

Problem Is C a theorem of P

Reference UVG

Hint Reduce PATH Problem A to LPQ Let X R S T b e an instance of PATH

Without loss of generality assume S fsgLettY b e an EDB relation sp ecifying that

no de Y is in T Let r U V W b e an EDB relation sp ecifying that the triple of no des

U V W isin R The basic logic program consists of the two rules aY tY and

aY r Y V W aV aW The relation a mo dels admissibilityin the path system

so as is a theorem of P if and only if s is admissible in the path system

Remarks Remains P complete even for very restricted programs In NC for programs

with the p olynomial fringe prop ertySee UV G for details See also AP for related

results

A Formal Languages

A ContextFree Grammar Memb ership CFGmem

Given A contextfree grammar G N T P S and a string x T

Problem Is x LG

Reference JL

Hint Reduce GEN to CFGmem Let W Vw b e an instance of GEN Construct the

grammar G W fagPw where P fx yz j y z xg fx j x V g It follows

that LG if and only if w is generated by V

Remarks Goldschlager remarks it is the presence of pro ductions in the input grammar

that make the memb ership question dicult Gol Lewis Stearns and Hartmanis log n

space algorithm LSH and Ruzzos AC hence NC algorithm Ruz for general

context free language recognition can b oth b e mo died to work with an free grammar

given as part of the input

A ContextFree Grammar EmptyCFGempty

Given A contextfree grammar G N T P S

Problem Is LGisempty

Reference JL Gol

Hint The reduction given in Problem A suces The following reduction of MCVP

to CFGempty due to Martin Tompa private communication is also of interest Given a

circuit construct the grammar G N T P Ssuch that N fi j v is a vertex in g

i

T fag and S nwherev is the output of Let g denote the value of gate g The

n

pro ductions in P are of the following form

For input v i a if v istrue

i i

A Comp endium of Problems Complete for P Preliminary RCS Revision

i jk if v v v

i j k

i j j k if v v v

i j k

Then v istrue if and only if i where fag

i

Remarks Note this reduction and the one for CFGinf havenopro ductions yet remain

complete The original pro of of Jones and Laaser reduced GEN to CFGempty Their pro of

used the reduction for CFGmem and instead checked if LG is empty JL

A ContextFree Grammar Innite CFGinf

Given Acontextfree grammar G N T P S

Problem Is LG is innite

Reference Gol JL

Hint Use a grammar similar to G in the pro of for CFGempty Problem A except

pro duction i a is replaced by i x and the pro ductions x a and x ax are also

added

A ContextFree Grammar Memb ership CFGmem

Given Acontextfree grammar G N T P S

Problem Is LG

Reference Gol JL

Hint Use a grammar similar to G in the pro of for CFGempty Problem A except

pro duction i a is replaced by i

A StraightLine Program Memb ership SLPmem

Given Astraightline program over alphab et jj with op erations taken from

ffg g and a string x

Problem Is x amemb er of the set constructed by the program

Reference Go o

Hint By noting there is a log space alternating Turing machine that parses x relative

to the program the problem is easily seen to b e in P Reduce MCVP to SLPmem by the

following true fg false and

Remarks The original reduction was from GEN Remains in P if is allowed The anal

ogous memb ership question for regular languages presented as regular expressions NFAs

is complete for NSPACElog n

A StraightLine Program Nonempty SLPnonempty

Given Astraightline program over alphab et jj with op erations taken from

ffg g and a string x

Problem Is the set constructed by the program nonempty

Reference Go o

Part I I P Complete Problems

Hint Reduce SLPmem to SLPnonempty Change nonempty constants to fg and test

memb ership of

Remarks With added SLPnonempty b ecomes complete for nondeterministic exp onen

tial time Go o

A Lab eled GAP LGAP

Given A xed context free language L over alphab et a directed graph G V E with

edges lab eled by strings in and twovertices s and t

Problem Is there a path from s to t such that the concatenation of its edge lab els is

in L

Reference Ruz GHR

Hint Reduce the wayDPDA acceptance problem Problem A to LGAP Let M

Q q Z b e a wayDPDA HU and let x b e an input string Without

loss of generality the PDA has a unique nal conguration q and accepts with empty stack

f

Z j Z gLetV b e the set with its head at the right end of the input Let f

containing the sp ecial vertex s together with all surface congurationsof the PD A ie

Q fjxjg There is an edge from hp ii to hq j i lab eled if and only if when

reading x the PDA has a move from p to q that moves its input head j i f g

i

Z Additionally there cells to the right p ops Z and pushes where

is an edge from the sp ecial vertex s to the initial surface conguration hq i lab eled Z

the initial stacksymb ol The designated vertex t is hq jxji Finally L LG where

f

G fS aS aS j a gfS g ie the semiDyck language D on jj letters Har

jj

Section

Remarks Remains P complete when L is D An equivalent statement is that it is P

complete to decide given a deterministic nite state automaton M whether D LM

If G is acyclic then the problem is complete for SAC LOGCFL Ruz

A TwoWayDPDA Acceptance DPDA

Given Atwoway deterministic pushdown automaton M and a string x

Problem Is x accepted by M

Reference Co oa Gal Gal Lad

Hint See eg HU for a denition of DPDAs Co ok Co oa gives a direct simulation

of a p olynomial time Turing machine by a logarithmic space auxiliary pushdown automaton

Galil Gal Gal shows existence of a P complete language accepted by a DPDA in

eect showing that the logarithmic space worktap e isnt crucial to Co oks simulation See

also Sud for a general reduction of auxiliary PDAs to ordinary PDAs Ladner Lad

gives a much more direct pro of by observing that a suitably enco ded version of CVP is

solvable byaDPDA basically by doing a depth rst search of the circuit using the stack

for backtracking

Remarks Remains in P when generalized to nondeterministic andor logarithmic space

auxiliary PDAs Co oa When restricted to p olynomial time PDAsoronewayPDAs

even with a logarithmic space worktap e the problem is in NC sp ecically it is complete

for LOGDCFL in the deterministic case and for LOGCFL SAC in the nondeterministic

A Comp endium of Problems Complete for P Preliminary RCS Revision

case

A Algebraic Problems

A Finite Horizon Markov Decision Pro cess FHMDP

Given A nonstationary Markov decision pro cess M S c p and an integer T Before

dening the problem we present some background on Markov decision pro cesses The term

nite horizon refers to the time b ound T S is a nite set of states and contains a desig

nated initial state s Lets denote the current state of the system for each time t

t

Asso ciated with each state s S is a nite set of decisions D A cost of cs i t is incurred

s

The next state s has probability distribution given at time t by making decision i D

s

t

by ps s it If c and p are indep endentoft then the pro cess is said to b e stationaryA

policy is a mapping that assigns to eachtimestept and eachstates a decision s t A

p olicy is stationary if is indep endent of time and can then b e suplied as an input

P

T

Problem Is the exp ectation of the cost cs s tt equal to

t t

t

Reference PT

Hint There is a p olynomial time algorithm for the problem that uses dynamic program

ming The reduction is from the monotone circuit value problem Problem A Let

c a b c i k denote an enco ding of MCVPwhere a denotes gate typ e and

i i i i

b and c denote the numb ers of gate is inputs A stationary Markov pro cess M S c p

i i

is constructed from the circuit instance as follows S has one state q for each i i k

i

There is an additional state in S called q Ifa b c corresp onds to a circuit input then

i i i

the corresp onding state q has a single decision with pq q and cost cq

i i i

if a is a false true input All other costs of this pro cess are There is one decision

i

for state q and pq q If a is an or gate then there are two decisions and

i

from state q The asso ciated probabilities are pq q and pq q The

i i b i c

i i

asso ciated costs are b oth If a is an and gate then there are two decisions and from

i

state q pq q and pq q with asso ciated costs The initial state

i i b i c

i i

is k corresp onding to the output gate of the circuit the time horizon T is k It is easy to

verify that the exp ected cost of the pro cess is if and only if the circuit evaluates to true

Remarks The reduction shows that the nite horizon stationary version of the problem is

P hard This problem is not known to b e in P Thedeterministic version of the FHMDP

which requires that p has only values or is in NC PT Note this last result holds

for b oth the stationary and nonstationary versions of the problem

A Discounted Markov Decision Pro cess DMDP

Given A stationary Markov decision pro cess M S c p and a real number

See Problem A for denitions This problem is innite horizon that is there is no time

b ound

P

t

Problem Is the exp ectation of the cost cs s t equal to

t t

t

Reference PT

Hint The problem can b e phrased as a linear programming problem and solved in p oly

nomial time The same construction as used in Problem A can b e used to showthis

Part I I P Complete Problems

problem is P complete

Remarks The innite horizon discounted deterministic problemisinNC PT The

deterministic problem requires that p has values only or

A Average Cost Markov Decision Pro cess ACMDP

Given A stationary Markov decision pro cess M S c p See Problem A for deni

tions This problem is innite horizon that is there is no time b ound

P

T

Problem Is the lim cs s tT

T t t

i

Reference PT

Hint The problem can b e phrased as a linear programming problem and solved in p oly

nomial time The reduction is from a synchronous variantofMCVP Problem A The

construction is a mo dication to that given in Problem A Instead of having states

corresp onding to circuit inputs going to a new state q they have transitions to the initial

state The limit is if and only if the circuit instance evaluates to true

Remarks The innite horizon average cost deterministic problem is in NC PT The

deterministic problem requires that p has values only or

A Gaussian Elimination with Partial Pivoting GEPP

Given An n n matrix A with entries over the reals or rationals and an integer l

th

Problem Is the pivot value for the l column p ositive when Gaussian elimination with

partial pivoting is p erformed on APartial pivoting is a technique used to obtain numerical

stability in whichrows of the matrix are exchanged so that the largest value in a given

column can b e used to p erform the elimination

Reference Vav

Hint The standard Gaussian elimination algorithm requires O n op erations Since the

size of the numb ers involved can b e b ounded by a p olynomial in n see Vav the problem

is in P Toshow completeness reduce NAND CVP to GEPP Without loss of generality

assume the inputs and gates of the circuit are numb ered in top ological order from to G

where G numb ers the output gate A G G matrix A a is constructed from the

ij

instance of CVP The entries of A are describ ed b elow A true circuit input i contributes

entry in p osition a and entry in p osition a For false input i or nand gate

ii ii

i A has entry in p osition a and in p osition a Ifgatei is an input to gate

ii ii

k then A has entry in p osition a and entry in p osition a For i G A

k i ki

has entry a All unsp ecied matrix entries havevalue The pivot value used

iGi

in eliminating column G is p ositive negative if and only if the circuit evaluates to false

true

Remarks The reduction do es not rely on large numb ers and therefore it shows that the

problem is strongly P complete Another decision problem that is strongly complete for

P based on Gaussian elimination with partial pivoting is as follows given matrix A and

th th

integers i and j is the pivot used to eliminate the j column taken from the initial i

row Vavasis also shows that Gaussian elimination with complete pivoting is P complete

In complete pivoting b oth rows and columns are interchanged so that the largest remaining

A Comp endium of Problems Complete for P Preliminary RCS Revision

matrix entry can b e used as a pivot The reduction for complete pivoting do es not show

the problem is strongly P complete This question is op en

A Iterated Mo d IM

Given Integers a b b

n

Problem Is a mo d b mod b modb

n

Reference KR

Hint Reduce NAND CVP to IM Without loss of generality assume the gates are num

b ered in reverse top ological order from G down to where the output gate is numb ered

Let y y denote the inputs and Y f g denote the value of input y The input

r l l

wires to gate g are numbered g and g Let a b e a bit vectoroflengthG whose

th

j bit is Y if edge j is incident from input y and otherwise Let O represent the set of

l l g

outedge lab els from gate g For g Gwe construct mo duli b b as follows

G

X

g g j g

b and b

g g

j O

g

The output gate in the NAND CVP instance has value if and only if

a mo d b modb mod b

G

Remarks The p olynomial iterated mo d problem is the problem in which

axb xb xareunivariate p olynomials over a eld F and the question is to deter

n

mine if

axmod b x mo d b x modb x

n

This problem is in NC KR The pro of technique used to showIMis P complete can

b e mo died to showthe superincreasing knapsack problem is also P complete KR The

sup erincreasing knapsack problem is dened analogously to the knapsack problem GJ

P

i

with weights w w except for i n w w

n i j

j

A Generalized Word Problem GWP

fs s js S g Given Let S b e a nite set and F b e the free group generated by S LetS

denote the set of all nite words over S where s denotes the inverse of sLetS Let

U fu u g S where m N and let x S

m

Problem Is x hU i That is is x in the subgroup of F generated by U

Reference AMb Ste

Hint Stewart rep orted an error in the result contained in AMb The reduction

in AMb is a generic one from a normal form Turing machine However it reduces a Tur

ing machine computation to a version of GWP where S is a countably innite set Ste

Stewart shows using the Nielsen reduction algorithm that this problem is still in P Stew

art calls this P complete problem the generalized wordproblem for countablygeneratedfree

groups GWPC He shows that GWPC is log space reducible to GWP thus proving GWP

is P complete as well Ste

Part I I P Complete Problems

Remarks For a natural number k GWPCk and GWPC k are the generalized

word problems for nitelygenerated subgroups of countablygenerated free groups where all

words involved are of length exactly k and at most k resp ectivelyStewart shows GWPCk

and GWPCk areP complete for k Ste When k the problems are complete

for symmetric log space SL The wordproblem for a freegroup is to decide whether x

equals the emptyword in F This problem is solvable in deterministic log space LZ

A Subgroup ContainmentSC

fs s js S g Given Let S b e a nite set and F b e the free group generated by S Let S

denote the set of all nite words over S where s denotes the inverse of sLetS Let

where m p N U fu u gV fv v g S

m p

Problem Is the group generated by U a subgroup of the group generated by V

Reference AMb

Hint Avariant of the Nielsen reduction algorithm can b e used to show the problem is in

P The reduction is from the generalized word problem Problem A Observe that for

any x S and U S x hU i if and only if hxi is a subgroup of hU i

Remarks Since hU i is a subgroup of hV i if and only if hU V i is normal in hV iit

follows that the normal subgroup problem which is also in P is P complete The problem

of determining whether hU i is normal in hU xi is also P complete AMb

A Subgroup Equality SE

fs s js S g Given Let S b e a nite set and F b e the free group generated by S Let S

where s denotes the inverse of sLetS denote the set of all nite words over S Let

where m p N U fu u gV fv v g S

m p

Problem Is hU i hV i

Reference AMb

Hint Avariant of the Nielsen reduction algorithm can b e used to show the problem is in

P The reduction is from the subgroup containment problem Problem A Observe hU i

is a subgroup of hV i if and only if hU V i hV i

A Subgroup Finite Index SFI

fs s js S g Given Let S b e a nite set and F b e the free group generated by S Let S

where s denotes the inverse of sLetS Let denote the set of all nite words over S

U fu u gV fv v g S where m p N

m p

Problem Is hU i a subgroup of hV i with nite index in hV i The index of U in V is the

numb er of distinct right cosets of U in V

Reference AMb

Hint Avariant of the Nielsen reduction algorithm can b e used to show the problem is in

P The reduction is from the subgroup containment problem Problem A Note hU i is

The problem a subgroup of hV i if and only if hU V i has nite index in hV iLetx S

of determining whether hU i has nite index in hU xi is also P complete AMb

A Comp endium of Problems Complete for P Preliminary RCS Revision

A Group Indep endence GI

Given Let S b e a nite set and F b e the free group generated by S LetS fs s js S g

where s denotes the inverse of sLetS denote the set of all nite words over S Let

where m N U fu u g S

m

Problem Is U independent That is do es each x hU i have a unique freely reducible

representation A word w is freely reducible if it contains no segment of the form ss or

s s

Reference AMb

Hint Avariant of the Nielsen reduction algorithm can b e used to show the problem is in

P The reduction is from an arbitrary p olynomial time Turing machine AMb

A Group Rank GR

Given Let S b e a nite set and F b e the free group generated by S LetS fs s js S g

Let denote the set of all nite words over S where s denotes the inverse of sLetS

k N LetU fu u g S where m N

m

Problem Do es hU i have rank k The rank is the numb er of elements in a minimal

generating set

Reference AMb

Hint Avariant of the Nielsen reduction algorithm can b e used to show the problem is in

P The reduction is from the group indep endence problem Problem A Observe U is

indep endent if and only if hU i has rank the numb er of elements in U

A Group Isomorphism SI

fs s js S g Given Let S b e a nite set and F b e the free group generated by S LetS

Let denote the set of all nite words over S where s denotes the inverse of sLetS

U fu u gV fv v g S where m p N

m p

Problem Is hU i isomorphic to hV i

Reference AMb

Hint Avariant of the Nielsen reduction algorithm can b e used to show the problem is in P

The reduction is from the group indep endence problem Problem A U is indep endent

if and only if hU i and hfs s gi are isomorphic

jU j

A Group Induced Isomorphism GI I

fs s js S g Given Let S b e a nite set and F b e the free group generated by S LetS

where s denotes the inverse of sLetS denote the set of all nite words over S Let

where m p N U fu u gV fv v g S

m p

Problem Do es the mapping dened by u v for i m induce an isomor

i i

phism from hU i to hV i

Reference AMb

Hint Avariant of the Nielsen reduction algorithm can b e used to show the problem is in P

The reduction is from the group indep endence problem Problem A U is indep endent

if and only if as dened ab ove induces an isomorphism from hU i to hfs s gi

jU j

Part I I P Complete Problems

A Intersection of Cosets IC

Given Let S b e a nite set and F b e the free group generated by S Let S fs s js S g

Let denote the set of all nite words over S where s denotes the inverse of sLetS

x y S LetU fu u gV fv v g S wherem p N

m p

Problem Is hU ix y hV i nonempty

Reference AMc

Hint A p olynomial time algorithm for the problem is given in AMc The reduction is

from an arbitrary p olynomial time Turing machine AMc

Remarks The intersection of right cosets problem and the intersection of left cosets

problem are subproblems of the intersection of cosets problems and they are b oth P complete

as well For example the right coset problem is P complete since hU ix hV iy is nonempty

if and only if hU ixy ehV i is nonemptywheree denotes the emptyword

A Intersection of Subgroups IS

fs s js S g Given Let S b e a nite set and F b e the free group generated by S Let S

where s denotes the inverse of sLetS Lete denote the set of all nite words over S

denote the emptyword Let U fu u gV fv v g S where m p N

m p

Problem Is hU i hV i hei

Reference AMc

Hint A p olynomial time algorithm for the problem is given in AMc The reduction is

straightforward from the intersection of right cosets problem see Problem A

A Group Coset Equality GCE

fs s js S g Given Let S b e a nite set and F b e the free group generated by S Let S

where s denotes the inverse of sLetS denote the set of all nite words over S Let

x y S LetU fu u gV fv v g S wherem p N

m p

Problem Is hU ix y hV i

Reference AMc

Hint A p olynomial time algorithm for the problem is given in AMc The reduction is

from Problem A

Remarks The following three decision problems are also P complete equality of right

cosets DoeshU ix hV iy equivalenceofcosets Are there x y such that hU ix

y hV i and equivalence of right cosets Are there x y such that hU ix hV iy AMc

A Conjugate Subgroups CS

Given Let S b e a nite set and F b e the free group generated by S Let S fs s js S g

denote the set of all nite words over S where s denotes the inverse of sLetS Let

where m p N U fu u gV fv v g S

m p

Problem Is there an x S suchthatx hU ix hV i

Reference AMc

Hint A p olynomial time algorithm for the problem is given in AMc The reduction is

A Comp endium of Problems Complete for P Preliminary RCS Revision

from the equivalence of right cosets problem see Problem A

Remarks The problem of determining whether x hU ix is a subgroup of hV i is also

P complete AMc

A Uniform Word Problem for Finitely Presented Algebras UWPFPA

Given A nitely presented algebra A M A and a pair of terms x y M is a nite set

of symbols and A M N denes the arity of eachsymbol N represents the nonnegative

integers M is partitioned into twosets G fa M j Aa g consists of generator

symbols and O fa M j Aa g consists of operator symbols The set of terms over

M is the smallest subset of M such that

all elements of G are terms

if is mary and x x are terms then x x isaterm

m m

Let denote the set of terms is a set of unordered pairs of terms called axioms is the

smallest congruence relation on satisfying the axioms of

Problem Is x y

Reference Koz

Hint A p olynomial time algorithm for the problem is given in Koz The reduction

is from the monotone circuit value problem Problem A Let B b e an instance MCVP

represented as a list of assignments to variables C C of the form C C C

n i i i

C C orC C C where i j k B is in MCVP provided valueC where n

j k i j k n

denotes the output gate of the circuit The reduction is as follows G fC C g

n

O f gBf

g B is in MCVP if and only if C

n

A Triviality Problem for Finitely Presented Algebras TPFPA

Given A nitely presented algebra A M A See Problem A for denitions

Problem Is A trivial That is do es it contain only one element

Reference Koz

Hint A p olynomial time algorithm for the problem is given in Koz The reduction

is from MCVP Construct as done in the pro of hint for Problem A Let

fC g Using notation from Problem A it follows that B is an instance of

n

MCVP if and only if C that is if and only if that is if and only if is

n

trivial

A Finitely Generated Subalgebra FGS

Given A nitely presented algebra A M A and terms x x y Let x

n

fy j x y g See Problem A for denitions

Problem Is y contained in the subalgebra generated byx x

n

Reference Koz

Part I I P Complete Problems

Hint A p olynomial time algorithm for the problem is given in Koz This problem is a

general formulation of GEN Problem A and so it follows that it is also P complete

A Finiteness Problem for Finitely Presented Algebras FPFPA

Given A nitely presented algebra A M A See Problem A for denitions

Problem Is A nite

Reference Koz

Hint A p olynomial time algorithm for the problem is given in Koz The reduc

tion is from MCVP and is similar to that used in Problem A The algebra con

structed in that pro of is mo died as follows add another generator b to G and the axioms

fb b b b b b b b g to to obtain is

nite if is trivial and otherwise it is innite

A Uniform Word Problem for Lattices UWPL

Given Let E b e a set of equations and e e an equation Wepresent some preliminary

denitions b efore dening the problem A lattice isasetL with two binary op erations

f g that satises the lattice axiomsLetx y z L The lattice axioms are as follows

asso ciativity x y z x y z x y z x y z

commutativity x y y x x y y x

idemp otence x x x x x xand

absorption x x y x x x y x

Let U b e a countably innite set of symbols The set of terms over U W U is dened

inductively as follows

If is in U then is in W U

If p q are in W U then p q p q are in W U

Let e and e b e terms over U Anequation isaformula of the form e e Avaluation

for a given lattice L is a mapping UL The valuation is extended to W U by

dening p q pq and p q p q A lattice satises an equation

e e under a valuation denoted L j e e if and only if e e A lattice

L satises a set of equations E denoted L j E if and only if L satises every member of

E under E implies e e denoted E j e e if and only if for every lattice L and

valuation such that L j E it follows that L j e e

Problem Do es E j e e

Reference Cos

Hint A p olynomial time algorithm for the problem is given in Cos The reduction

is from the implication problem for propositional Horn clauses JL See Problems A

and A Let b e a set of prop ositional formulas of the form x x x where x x

i j k

A Comp endium of Problems Complete for P Preliminary RCS Revision

are prop ositional variables Let b e the formula x x x The problem is to test if

implies Let represent the formula x x x In the instance of UWPL we construct

j j k

the equation as follows LetE f j g It follows that

i j k i k

implies if and only if E j

Remarks The problem remains P complete if we use inequalities instead of equations

Furthermore the problem remains P complete when E and the terms are represented

by dags instead of trees However if E and the terms are represented as trees the

problem is in DLOG Cos This problem is called the identity problem for lattices

A Generator Problem for Lattices GPL

Given Let L b e a lattice Let E b e a set of equations and e g g be terms over

n

U Let UL be a valuation see Problem A We present some preliminary

denitions rst Let X LThesublattice generated by X is the smallest subset of L that

contains X and is closed under the op erations of L e is generatedby g g in L under

n

denoted L j gene g g if and only if e is in the sublattice of L generated

n

by the set fg j i ng E implies that e is generatedby g g denoted E j

i n

geng g if and only if for every lattice L and valuation such that L j E it

n

follows that L j gene g g

n

Problem Do es E j gene g g

n

Reference Cos

Hint A p olynomial time algorithm for the problem is given in Cos The reduction is a

continuation of the reduction used in Problem A Since E j if and only if E j

gen it follows that GPL is also P complete

Remarks The problem remains P complete when E and the terms are represented

by dags instead of trees However if E and the terms are represented as trees the

problem is in DLOG Cos This problem is called the generator problem for free lattices

A Bo olean Recurrence Equation BRE

Given M B F j where M is an m n Bo olean matrix B is an n n Bo olean matrix

F is an n Bo olean vector and j is an integer in the range to n

Problem Is the rst entry of M Y a Y is dened recursively as Y F Y B Y

j j k k

for k

Reference BBMS

Hint The reduction is from an alternating Turing machine that uses O log n space

k k

Remarks If j log n then the problem is complete for AC BBMS That is the

k

class of problems accepted by alternating Turing machines in O log n space and O log n

alternations

A Geometry

A Plane Sweep Triangulation PST

Given An n vertex p olygon P that maycontain holes and a designated vertex u

Part I I P Complete Problems

Problem Is a vertical edge connecting to u in the plane sweep triangulation of P The

plane sweep triangulation is the triangulation pro duced bysweeping from top to b ottom a

horizontal line L When L encounters a vertex v of P each diagonal from v to another vertex

in P which do es not cross a previously drawn diagonal is added to the triangulation

Reference ACG

Hint It is easy to see the plane sweep triangulation algorithm runs in p olynomial time

The reduction is from a variant ofplanarCVP Problem A The new version of PCVP

consists of not gates of fanout or gates of fanout routing gates and fanout gates

that take one value and pro duce two copies of it This instance of PCVP is required to b e

laid out on a planar grid in a sp ecial manner with alternating layers of routing and logic

The reduction involves constructing geometricgadgets for routing left and righ t shifts

in the grid verticalwires fanout or gates and not gates The presence absence of

avertical edge in the triangulation denotes a true falsevalue The vertex u is a sp ecial

targetv ertex in the output gate of the circuit A vertical line is connected to u in the

triangulation if and only if the circuit evaluates to true

Remarks The problem of nding some arbitrary triangulation is in NC Go o If

the p olygon P is not allowed to have holes then the complexity of the problem is op en

In ACG they conjecture that this restricted version is in NC

A Oriented Weighted Planar Partitioning OWPP

Given A set of nonintersecting line segments s s in the plane a set of asso ciated

n

integer weights w w and two designated segments r and t The segments are

n

oriented meaning that there are only three dierent p ossible slop es for the segments

Problem Do segments r and t touchin the partitioning of the plane constructed b y

extending segments in the order of their weights Segments are extended until they reach

another segment or a previous segment extension

Reference ACG

Hint It is easy to see that the pro cess of extending the segments can b e p erformed in

p olynomial time The reduction is from the same version of PCVP as used in Problem

A The gates of the instance of PCVP are numb ered in top ological order Gadgets are

constructed for routing and for logic There are gadgets for right and left shifts fanout

gates or gates not gates and true inputs true values in the circuit are transmitted as

vertical extensions of segments The most interesting gadget is the one for the not gate and

we describ e this The gadgets for the other constructs all involve only two dierent slop ed

segments whereas to simulate not gates three dierent slop es are required The instance of

PCVP was laid out on a grid so we consider a not gate numb ered i that receives its input on

channel j and has its output on channel k Ablocking segment is one that is used to prevent

the extension of another line segmentandwhoseweightisvery large These segments dont

play an active role in simulating the gate The not gadget consists of six blo cking segments

and three additional segments called onet woand three indicating their relativ e

weights That is segment one is pro cessed rst within the gadget followed bytwo and then

three Segmenttwo is a horizontal segment whose p otential extension spans channels j and

k and is blo cked on b oth ends Two lies directly to the left of channel j Segmentthree

isavertical segmentonchannel k blo cked ab ovesegmen ttwo but with the p ossibilityof

A Comp endium of Problems Complete for P Preliminary RCS Revision

b eing extended downward across twos extension Channel j the input to not gate iis

blo cked ab ove segmenttwo Segment one has a slop e of and its p otential extension is

blo cked at b oth ends Segment one lies completely to the left of channel j Its rightward

extension would cross channel j as well as segmenttwos extension Wenow describ e how

this gadget simulates a not gate If the input to gate i is true false then the vertical

segmentonchannel j has has not b een extended to where it is blo cked This prevents

allows segment one from b eing extended across segmenttwo Thus segmenttwocanbe

cannot b e extended towards the right across channel k This prevents allows segment

three from b eing extended across segmenttwos extension indicating a false truevalue

for the output of the gate The assignments of weights and the construction of all gadgets

can b e accomplished in log space Segment r is the vertical segment corresp onding to the

output wire of the circuit Segment t is a sp ecial output pad r will touch t when extended

if and only if the circuit evaluates to true

Remarks The complexity of the oriented version of the problem is op en ACG

In ACG they remark that the problem has b een reduced to an instance of MCVP that

has a very restricted top ology although not planar Thus it is op en whether or not this

version of the problem is in NC

A VisibilityLayers VL

Given Asetofn nonintersecting line segments in the plane and a designated segment s

Problem Is the lab el assigned to segment s bythe visibility layering process congruent

to one mo d three The visibilitylayering pro cess is rep eatedly to compute and delete the

upper envelope of the remaining set of segments and lab el those segments with the current

depth The upp er envelop e consists of those segments visible from the p oint A

segment is visible from a p oint p if a ray cast from p can hit the segment b efore hitting any

other segment

Reference ACG Her

Hint The visibilitylayering pro cess is p olynomial time Her The reduction presented

in Her is from the monotone CVP Wesketch this reduction The gates in the instance of

CVP are assumed to b e numb ered in top ological order A grid is constructed that consists

of V rows and E columns where V E is the numb er of no des edges in the dag

corresp onding to the circuit Gadgets are constructed for the and and or gates Gadgets

consist of horizontal line segments of varying lengths The gadget for and gate k with

inputs i and j ij and outputs l and m lm consists of three horizontal segments

situated in row k of the grid One segment spans column i one segment spans column j and

another segment spans from column i through column m The gadget for or gate k with

inputs i and j ij and outputs l and m lm consists of three horizontal segments

situated in row k of the grid One segment spans column j one segment spans columns i

through j and another segment spans columns j through m If an input asso ciated with a

given column is true false then a horizontal segment is put is not put in to span that

column in row Deep enerswhic h are horizontal line segments spanning single columns

of the grid are used to make sure gate input values arrive at the right timeand also to

make sure that once a gate has b een evaluated its outputs aect only the desired gates

The output of the circuit is true if and only if a s has a lab el whose value is congruentto

Part I I P Complete Problems

one mo d three

Remarks The reduction given in ACG is similar and is also from a variantofMCVP

The main dierence is in the way fanout is treated The version of MCVP used in ACG

consists of crossing fanout gates single output and gates and single output or gates An

instance consists of alternate routing and logic layers Gadgets are constructed for the three

typ es of gates and a similar decision problem to the one in Her is p osed to determine

the output of the circuit If the length of all segments is required to b e the same then the

complexity of the problem is not known ACG In ACG they conjecture that this

version of the problem is in NC

A Real Analysis

A Real Analogue to CVP RealCVP

Given A feasible real function V dened on A real function f is feasible if

n n

given a suciently accurate xedp oint binary approximation to x a xedp oint

n O

binary approximation to f x with absolute error can b e computed in time n

O

n

Suciently accurate means that the error in approximating the input x is This

xes the numb er of input bits the continuityoff limits its range and thus xes the number

of output bits b oth are p olynomial in n

n

Problem Compute V x with absolute error

Reference Ho o Ho o

Hint The function V computes the continuous analog of the circuit value function by

mapping circuit descriptions along with their p ossible inputs onto the real line Toevalu

as an integer and the bits x as an nbit ate the circuit on input x treat the enco ding

xedp oint binary fraction and add the two The value of V x is then a rational number

that enco des the values of the gates of on the input xTomake V continuous b etween

these evaluation p oints V is simply linearly interp olated

Remarks The same function yields a family of P complete p olynomials fp g computable

n

by feasiblesizemagnitude circuits A arithmetic circuit family is feasiblesize

th n n

magnitude if the n memb er is p olynomial size and its output over the interval

can b e computed without generating anyintermediate values with magnitude exceeding

O

n

A Fixed Points of Contraction Mappings FPCM

Given An NC real function C that b ehaves as a contractor on some interval I with

integer endp oints contained in The end p oints of I are sp ecied as integers

n

A real function f is in NC if an approximation to f x with absolute error for

n n

x can b e computed in NC with the same inputoutput conventions as for

RealCVP

n

Problem Compute the xed p ointofC in I with absolute error

Reference Ho o

Hint Same basic technique of RealCVP but the function C evaluates the circuit level by

level thus converging to a xed p oint which enco des the nal state of the circuit Finding

A Comp endium of Problems Complete for P Preliminary RCS Revision

the xed p ointisinP since each iteration of the contraction mapping reduces the width of

the interval by some constant factor

Remarks This provides an argument that fast numerical metho ds based on xed p oints

probably have to use contraction maps with b etter than linear rates of convergence such

as Newtons metho d

A Inverting a OnetoOne Real Functions IOORF

Given An NC real function f on that is increasing and has the prop erty that

f f Thus there is a unique ro ot x such that f x A real function f is

n n n

in NC if an approximation to f x with error forx can b e computed

in NC

n

Problem Compute x with error

Reference Ko

Hint Map intermediate congurations of a log space DTM onto the real line

Remarks This problem was expressed originally in terms of log space computability and

reductions if f is log space computable then f is not log space computable unless

DLOG P The problem remains hard even if f is required to b e dierentiable

A Miscellaneous

A General Deadlo ck Detection GDD

Given Amultigraph D V E with V where fp p g is the set of

n

process nodes and fr r g is the set of resourcenodes and a set T ft t g

m m

where t denotes the numb er of units of r The bipartite multigraph D represents the state

i i

of the system The edges in V are of the form p r denoting a request of pro cess p for

i j i

resource r or of the form r p denoting an allo cation of resource r to pro cess p

j j i j i

Problem Is D a dead lock state A deadlo ck state is one in which there is a nontrivial

subset of the pro cesses that cannot change state and will never change state in the future

Reference Spi

Hint The reduction is from MCVP Let denote an instance of MCVP

n

There will b e one pro cess p asso ciated with each and one additional sp ecial pro cess p

i i

The representation of gates is as follows

If is an input with a false truevalue then the edge p rf rt p is in D

i i i i i

and where j k i i the and gate has fanout Then Supp ose

j k i i

edges p r p r r p and r p are added to D for and edges

i i j i i k i j j i k k i

p r p r r p and r p are added to D for

i i j i i k i j j i k k i

Supp ose or where j k i i the or gate has fanout

i i j k

Then edges p r r p and r p are added to D for and edges

i i jk i jk j i jk k i

p r r p and r p are added to D for

i i jk i jk j i jk k i

Part I I P Complete Problems

Edges are also added for the sp ecial pro cess p as follows for i n add rf p

i

and p rt and for j k n add p r and p r The graph D is not in a

i nj nj k

deadlo ck state if and only if is true

n

Remarks Notice in the reduction the maximum numb er of units of any resource is two

The problem is in NC if t for all i Spi That is if there is only one unit of each

i

resource If the system states are expedient the resource allo cator satises them as so on as

p ossible and at most one request can b e connected to any pro cess at a given time then the

problem is in NC Spi

A Two Player Game GAME

Given Atwo player game G P P W sM dened by P P W P P

s P and M P P P P P is the set of p ositions in which it is player is turn

i

to move W is the set of immediate winning positions dened b elow for player one and

s is the starting p osition M is the set of allowable moves if p q M and p P or

P then player one or two maymove from p osition p to p osition q in a single step A

p osition x is winning for player one if and only if x W orx P and x y M for

some winning p osition y orx P and y is winning for every movex y in M

Problem Is s a winning p osition for the rst player

Reference JL

Hint Reduce AMCVP to GAME or gates in the circuit corresp ond to winning p ositions

for player one and gates to winning p ositions for player two W is the set of all inputs

having value true and s is the output s is winning if and only if the output of the circuit

is true

Remarks The original reduction by Jones and Laaser was from GEN to GAME JL

Given W V and w construct the game G WW WVwM where the allowable

moves are given by

M fp q r j q r pg fp q p j p q W gfp q q j p q W g

Player one attempts to prove that a no de p is generated by V They do this by exhibiting

two elements q and r also claimed to b e generated by V such that p q r Player two

attempts to exhibit an element of the pair that is not generated by V Since GAME is an

instance of ANDOR graph solvability it follows that determining whether an ANDOR

graph has a solution is also P complete Kas

A Cat Mouse CM

Given A directed graph G V E with three distinguished no des c m and g

Problem Do es the mouse haveawinning strategy in the game The game is played as

follows The cat starts on no de c the mouse on no de m and g represents the goal no de

The cat and mouse alternate moves with the mouse moving rst Eachmove consists of

following a directed edge in the graph Either player has the option to pass by remaining

on the same no de The cat is not allowed to o ccupy the goal no de The mouse wins if it

reaches the goal no de without b eing caught The cat wins if the mouse and cat o ccupythe

A Comp endium of Problems Complete for P Preliminary RCS Revision

same no de

Reference CSa Sto

Hint The reduction is from a log space alternating Turing machine M Assume that M

starts in an existential conguration I has a unique accepting conguration A that is exis

tential and each existential universal conguration has exactly two immediate successors

b oth of which are universal existential A directed graph is constructed with the number

of no des in the graph prop ortional to the numb er of congurations of M We illustrate only

how existential congurations can b e simulated and do not account for the cat or mouse

b eing able to pass on a move There are twocopiesofeach conguration C denoted C

and C in the graph we construct The graph has additional no des as well Consider an

existential conguration X of M with its two succeeding universal congurations Y and Z

Assume that the cat is on X the mouse is on X and it is the mouses turn to move X

is directly connected to Y and Z whereas X is connected to twointermediate no des Y

and Y Y Z is connected to Y Z and Y Z Both Y and Z are connected to g

The mouse simulates an existential moveof M bymoving to either Y of Z If the mouse

moves to Y Z then the cat must moveto Y Z Otherwise the mouses next move can

be to Y Z and from there onto g uncontested From Y Z the mouse must moveto

Y Z and an universal move is ready to b e simulated The simulation of universal moves

is fairly similar with the cat moving rst The game starts with the cat on I and the mouse

on I There is an edge from A to g M will accept its input if and only if the mouse has

a winning strategy

A Longcake LONGCAKE

Given Two players H and V a token and an m n Bo olean matrix M We describ e

how the game is played Initially the token is on placed on p osition m of M itisH s

turn to move and the current submatrix is M The term current submatrix denotes the

p ortion of M that the game is currently b eing played on H s turn consists of moving the

token horizontally within the current submatrix to some entry m At this p oint

ij

either all columns to the left of j or all columns to the rightofj are removed from the

current submatrix dep ending on which causes fewer columns to b e removed Notice the

token o ccupies a corner of the current submatrix again V s turn is similar except V moves

vertically and rows are removed The rst player with no moves left loses

Problem Do es H have a winning strategy on M

Reference CT

Hint The reduction is p erformed in two stages First MCVP is reduced to an acyclic

version of Schaefers Geography game SchcalledACYCLICGEO This proves that

ACYCLICGEO is P complete Second ACYCLICGEO is reduced to LONGCAKE

Remarks The game SHORTCAKE is the same as LONGCAKE except the larger p ortion

of the current submatrix is thrown awaySHORTCAKE is complete for AC CT An

other variant of these games called SEMICAKE is complete for LOGCFL SAC CT

Part I I Op en Problems

B Op en Problems

This section contains a list of op en problems The goal is to nd an NC algorithm or a P

completeness pro of for the problem Many of these op en questions are stated as computation

problems to b e as general as p ossible The problems listed are divided into the following

categories algebraic problems graph theoretic problems and miscellaneous problems

B Algebraic Problems

B Integer Greatest Common Divisor IntegerGCD

Given Two nbit p ositiveintegers a and b

Problem Compute gcd a b

Reference Gat

th

Remarks For n degree p olynomials p q Qx computing gcdp q isin NC via an

NC reduction to determinant CSb BCP IntegerGCD is NC reducible to short

vectors dimension Problem B Gat

B Extended Euclidean Algorithm ExtendedGCD

Given Two nbit p ositiveintegers a and b

Problem Compute integers s and t such that as bt gcda b

Reference BGH

th

Remarks The analogous problem for n degree p olynomials is in NC BGH

B Relative Primeness RelPrime

Given Two nbit p ositiveintegers a and b

Problem Are a and b relatively prime

Reference Tom

Remarks Is a sp ecial case of IntegerGCD Problem B

B Mo dular Inversion Mo dInverse

Given An nbit prime p and an nbit p ositiveinteger a such that p do es not divide a

Problem Compute b such that ab mo d p

Reference Tom

Remarks Mo dInverse is reducible to ExtendedGCD compute s t such that as pt

p

gcda p and also to Mo dPower even restricted to prime mo duli compute a mo d p

and apply Fermats Little Theorem

B Mo dular Powering Mo dPower

Given Positive nbit integers a bandc

b

Problem Compute a mo d c

A Comp endium of Problems Complete for P Preliminary RCS Revision

Reference Co o

Remarks The complexity of the problem is op en even for nding a single bit of the

desired output The problem is in NC for smo oth c that is for c having only small

prime factors Gat In the interesting sp ecial case when c is a prime the antithesis

of smo oth the problem is op en Mo dInverse Problem B is reducible to this restriced

th

case The analogous problems when a and c are n degree p olynomials over a nite eld

of small characteristic modular polynomial exponentiation and polynomial exponentiation

are in NC FT They are op en for nite elds having large sup erp olynomial charac

teristic Note that Mo dPower can b e reduced to the p olynomial version with exp onential

characteristic simply by considering degree p olynomials

B Short Vectors SV

n

Given Input vectors a a Z that are linearly indep endentover Q

n

P

n

Problem Find a nonzero vector x in the Z mo dule or lattice M a Z Z such

i

o n

P

n

is the L norm where ky k y that kxk ky k for all y M

i

Reference Gat

Remarks Lenstra Lenstra and Lovasz LLL show that the problem is in P Inte

gerGCD Problem B is NC reducible to SV Gat

B Short Vectors Dimension SV

Given Input vectors a a Z that are linearly indep endentover Q and an arbitrary

number c

n o

Problem Find a nonzero vector x such that kxk cky k for all y M

Reference Gat

Remarks IntegerGCD Problem B is NC reducible to SV Gat

B Sylow Subgroups SylowSub

Given A group G

Problem Find the Sylow subgroups of G

Reference BSL

Remarks The problem is known to b e in P Kan however the NC question is op en

even for solvable groups BSL For a p ermutation group G testing memb ership in G

nding the order of G nding the center of G and nding a comp osition series of G are all

known to b e in NC BSL Babai Seres and Luks presentseveral other op en questions

involving group theory BSL

B TwoVariable Linear Programming TVLP

Given A linear system of inequalities Ax b over the rationals where eachrowof A has

at most two nonzero elements

Problem Find a feasible solution if one exists

Reference LMR

Part I I Op en Problems

k

log n

Remarks There is a known p olylog algorithm that uses n pro cessors on a CREW

PRAM LMR

B Univariate Polynomial Factorization over Q UPFQ

th

Given An n degree p olynomial p Qx

Problem Compute the factorization of p over Q

Reference Gat

Remarks UPFQisNC reducible to short vectors Problem B Gat

B Graph Theory Problems

B EdgeWeighted Matching EWM

Given A graph G with p ositiveinteger weights on its edges

Problem Find a matching of maximum weight

Reference KUW

Remarks EWM is in RNC but not known to b e in NC if all weights are p olynomially

b ounded KUW LFMM Problem B and hence all of CC is NC reducible to

ranke

EWM by assigning weight to edge e

B Bounded Degree Graph Isomorphism BDGI

Given Two graphs G and H The vertices in G and H have maximum degree at most k

a constant indep endent of the sizes of G and H

Problem Are G and H isomorphic

Remarks Luks showed the problem is in P FHL Without the degree b ound the

problem is in NP but not known to b e in P nor is it known to b e either P hard or NP

complete

B Graph Closure GC

Given An undirected graph G V E and a designated edge e u v

Problem Is e in the closure of G That is the graph obtained from G by rep eatedly

joining nonadjacent pairs of vertices u and v whose degree sum is at least jV j

Reference Khu

Remarks With the following mo dication the problem b ecomes P complete Problem

A add a set of designated edges E such that only vertices whose degree sum is at

least jV j and whose corresp onding edge is in E may b e added to the closure Khu

B Restricted Lexicographically First Indep endent Set RLFMIS

Given A planar bipartite graph G withanumb ering on the vertices

Problem Find the lexicographically rst maximal indep endent set

Reference Miy

A Comp endium of Problems Complete for P Preliminary RCS Revision

Remarks See Problem A Finding the lexicographically rst maximal subgraph

of maximum degree one in planar bipartite graphs of degree at most three is P

complete Miy

B Maximal Path MP

Given A graph G with a numb ering on the vertices and a designated vertex r

Problem Find a maximal path originating from r That is a path that cannot b e extended

without encountering a no de already on the path

Reference AMb

Remarks The lexicographically rst maximal path problem Problem A is P complete

even when restricted to planar graphs with maximum degree three If the maximum degree

of anyvertex in G is at most then there is an algorithm that can nd a maximal path in

O log n time using n pro cessors AMb There is also an NC algorithm for nding

a maximal path in planar graphs AMb

B Maximal Indep endent Set Hyp ergraph MISH

Given A nite collection C fC C g of nite sets

n

Problem Find a sub collection D D D of pairwise disjoint members of C such

k

that for every set S C D there is a T D such that S T

Reference BL KR

Remarks If C is a collection of two element sets dimension then the problem

b ecomes the maximal indep endent set problem which is known to b e in NC see Problem

A Beame and Luby giveanRNC algorithm for dimension O as well as an algorithm

for the general problem that is conjectured to b e RNC

B Geometric Problems

B Oriented Weighted Planar Partitioning OWPP

Given A set of nonintersecting segments s s in the plane a set of asso ciated inte

n

ger weights w w and two designated segments r and t The segments are oriented

n

meaning that there are only two dierent p ossible slop es for the segments

Problem Do segments r and t touchin the partitioning of the plane constructed b y

extending segments in the order of their weights Segments are extended until they reach

another segment or a previous segment extension

Reference ACG

Remarks The oriented version Problem A in which three dierent slop es are al

lowed is P complete ACG

B Limited Reection RayTracing LRRT

Given A set of n at mirrors of lengths l l their placements at rational p oints in the

n

plane a source p oint S the tra jectory of a single b eam emitted from S and a designated

Part I I Op en Problems

mirror M

Problem Determine if M is hit by the b eam within n reections At the mirrors the angle

of incident of the b eam equals the angle of reection

Reference GHR

Remarks The general ray tracing problem is to determine if a mirror is ever hit bythe

b eam When the mirrors are p oints that is have no length the general problem is in

NC dlTG In two or more dimensions the general problem is in PSPACE Inthree

dimensions with mirrors placed at rational p oints the general problem is PSPACE hard

The general problem is op en for all mirrors of a xed size as well See RTY for a detailed

discussion

B Restricted Plane Sweep Triangulation SWEEP

Given An n vertex p olygon P without holes

Problem Find the triangulation computed bytheplane sweep triangulation algorithm

Reference ACG

Remarks The problem of nding some arbitrary triangulation is in NC Go o If the

p olygon is allowed to have holes then the problem is P complete ACG See Problem

A

B Successive Convex Hulls SCH

Given A set S of n p oints in dimension d and a designated p oint p

th

Problem Determine if p is in the i remaining convex hull that is formed by rep eatedly

nding and removing convex hulls from S

Reference ChaAta Cha

Remarks The problem is op en for two dimensions and up

B Unit Length VisibilityLayers ULVL

Given Asetofn unit length horizontal nonintersecting line segments in the plane a

designated segment sandaninteger d

Problem Is the lab el assigned to segment s by the visibility layering process d The

visibilitylayering pro cess is rep eatedly to compute and delete the upper envelope of the

remaining set of segments and lab el those segments with the current depth The upp er

envelop e consists of those segments visible from the p oint

Reference ACG

Remarks The problem is P complete if the restriction on unit lengths is removed ACG

See Problem A

B CC Problems

Several reseachers indep endently suggested lo oking at the complexity of the Comparator

Circuit Value Problem or CCVP Problem B CC is the class of problems reducible

to CCVP Mayr and Subramanian MS Problems in this section are all equivalent to

A Comp endium of Problems Complete for P Preliminary RCS Revision

ie NC interreducible with CCVP While the evidence is less comp elling than that for

RNC problems Section B it is generally considered unlikely that these problems are

P complete b ecause of the lack of fanout in comparator circuits On the other hand no fast

p

O

n log n algorithms parallel algorithms are known for them with the partial exception of

for CCVP and some related problems Such algorithms were indep endently discovered by

D Soroker unpublished and byMayr and Subramanian MS Richard Anderson also

p

unpublished has improved the algorithms so as to use only n pro cessors Mayr and

Subramanian note that these algorithms are P complete in the sense of Section It is

known that NL CC P MS

Most of the results describ ed in this subsection come from MS See also Sub

Sub Fed

B Comparator Circuit Value Problem CCVP

Given An enco ding of a circuit comp osed of comparator gates plus inputs x x

n

A comparator gate outputs the minimum of its two inputs on its rst output wire and

outputs the maximum of its two inputs on its second output wire The gate is further

restricted so that each output has fanout at most one

Problem Do es on input x x output

n

Reference Co o MS

Remarks Co ok shows that CCVP is NC equivalent to computing the lexicographically

rst maximal matching in a bipartite graph Mayr and Subramanian MS show that this

problem is NC equivalent to stable marriage Problem B

B Lexicographically First Maximal Matching LFMM

Given A graph G with an ordering on its edges

Problem Find the lexicographically rst maximal matching

Reference Co o MS

Remarks LFMM is NC equivalent to Comparator CVP Problem B MS This

problem resembles the lexicographically rst maximal indep endent set problem Problem

A known to b e P complete A P completeness pro of for LFMM would imply that

edgeweighted matching Problem B is also P complete

B Stable Marriage SM

Given For each memb er of a communityofn men and n women a ranking of the opp osite

sex according to their preference for a marriage partner

Problem Find n marriages such that the set of marriages is stable The set is unstable

if a man and woman exist who are not married to each other but prefer each other to their

actual mates

Reference MS

Remarks See eg Gib for background on the Stable Marriage problem SM is

NC equivalent to Comparator CVP Problem B MS Several variations on Stable

Marriage are also known to b e equivalent to CCVP see MS

Part I I Op en Problems

B RNC Problems

The problems in this subsection are all known to b e in RNC or FRNC but not known to

be in NC A pro of that any of them is P complete would b e almost as unexp ected as a

pro of that NC P See Section for more discussion

B Directed or Undirected Depth First SearchDFS

Given A graph G and a vertex s

Problem Construct the depth rst searchnumb ering of G starting from vertex s

Reference AA AAK

Remarks RNC algorithms are now known for b oth the undirected AA and di

rected AAK cases subsuming earlier RNC results for planar graphs Smi For DAGs

DFS is in NC Grea

B Maximum Flow MaxFlow

Given Directed graph G with each edge lab eled in unary with a capacity c and two

i

distinguished vertices source s and sink t

Problem Find a maximum ow

Reference Fea KUW

Remarks Feashows the problem of nding the value of the maximum owtobe

in RNC KUWshowhow to construct a maximum ow also in RNC Both problems

remain in RNC when capacities are p olynomially b ounded but are P complete when ca

pacities are arbitrary Problem A Venkateswaran Ven shows that MaxFlowis

hard for AC

B Maximum Matching MM

Given A graph G

Problem Find a maximum matching of GAmatching is a subset of the edges of G such

that no two edges of the subset are adjacentinG A matching is a maximum matching if

no matching of larger cardinality exists

Reference Fea KUW MVV

Remarks Feather Feashows that the problem of nding the size of maximum matching

is in RNC Karp Upfal and Wigderson KUWgave the rst RNC algorithm for nding

the maximum matching A more ecient algorithm was given by MulmuleyVazirani

and Vazirani MVV Karlo shows howany RNC algorithm for matching can b e made

errorless Kar Maximum edgeweighted matching for unary edge weights and maximum

vertexweighted matching for binary vertex weights are also known to b e in RNC KUW

MVV

B Perfect Matching Existence PME

Given A graph G

A Comp endium of Problems Complete for P Preliminary RCS Revision

Problem Do es G have a p erfect matching A perfect matching isamatching where each

vertex is incident to one edge in the matching

Reference KUW MVV

Remarks See remarks for Problem B PME seems to b e the simplest of the matching

problems not known to b e in NC

Part I I Op en Problems

C Undigested Pap ers

Listed b elow are a few pap ers weareaware of but have not yet added to the main problem

lists They will app ear in the nal version

KKS KS

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Problem Index

Problem Index

ContextFree Grammar

MaxFlow B

Memb ership A

Maximum Flow B

ContextFree Grammar Empty A

Oriented Weighted Planar

ContextFree Grammar Innite A

Partitioning B

ContextFree Grammar

DPDA A

Memb ership A

OWPP B

CS A

Oriented Weighted Planar

CVP A

Partitioning A

DFS B

OWPP A

Directed or Undirected Depth

ABFS A

First Search B

ACMDP A

Discounted Markov Decision

AGAP A

Pro cess A

Alternating Breadth First

DMDP A

Search A

Edge Maximal Acyclic

Alternating Graph Accessibili ty

Subgraph A

Problem A

EdgeWeighted Matching B

Alternating Monotone Fanout

EMAS A

CVP A

EWM B

AMCVP A

Extended Euclidean Algorithm B

ArithCVP A

ExtendedGCD B

Arithmetic Circuit Value

FFDBP A

Problem A

FGS A

Average Cost Markov Decision

FHMDP A

Pro cess A

Finite Horizon Markov Decision

BDGI B

Pro cess A

BDS A

Finitely Generated Subalgebra A

Bo olean Recurrence Equation A

Finiteness Problem for Finitely

Bounded Degree Graph

Presented Algebras A

Isomorphism B

First Fit Decreasing Bin

BreadthDepth Search A

Packing A

BRE A

Fixed Points of Contraction

Cat Mouse A

Mappings A

CCVP B

FPCM A

CFGmem A

FPFPA A

CFGempty A

GAME A

Gaussian Elimination with

CFGinf A

Partial Pivoting A

CFGmem A

GCE A

Circuit Value Problem A

GC B

CM A

Comparator Circuit Value

GDD A

Problem B

Generability A

General Deadlo ck Detection A Conjugate Subgroups A

A Comp endium of Problems Complete for P Preliminary RCS Revision

General Graph Closure A Lab eled GAP A

Lexicographically First

General List Scheduling A

Vertex Coloring A

Generalized Word Problem A

Lexicographically First

Generator Problem for Lattices A

Blo cking Flow A

GEN A

Lexicographically First Depth

GEPP A

First Search Ordering A

GGC A

Lexicographically First

GI I A

Maximal Clique A

Lexicographically First

GI A

Maximal Indep endent Set A

GLS A

Lexicographically First

GPL A

Maximal Matching B

Graph Closure B

Lexicographically First

Group Coset Equality A

Maximal Path A

Group Indep endence A

Lexicographically First

Group Induced Isomorphism A

Maximal Subgraph for A

Group Isomorphism A

LE A

Group Rank A

LFBF A

GR A

LFDFS A

GWP A

LFDVC A

HCS A

LFMC A

HDS A

LFMIS A

HF A

LFMM B

HGAP A

LFMP A

Hierarchical Graph

LFMS A

Accessibility Problem A

LGAP A

High Connectivity Subgraph A

Limited Reection RayTracing B

High Degree Subgraph A

Linear Equalities A

Homologous Flow A

Linear Inequalities A

Horn Unit Resolution A

Linear Programming A

HORN A

LI A

IC A

Lo cal Optimality

IM A KernighanLin Verication A

Integer Greatest Common

Logical Query Program A

Divisor B

LOKLV A

IntegerGCD B

LONGCAKE A

Intersection of Cosets A

Longcake A

Intersection of Subgroups A

LP A

Inverting a OnetoOne Real

LQP A

Functions A

LRRT B

Inverting an NC Permutation

MAXFLIP Verication A

A

MAXFLIPV A

InvNCPerm A

MaxFlow A

IOORF A

Maximal Indep endent Set

IS A

Hyp ergraph B

Iterated Mo d A Maximal Path B

Problem Index

Maximum Flow A Relaxed Consistent Lab eling A

Maximum Matching B

RelPrime B

Restricted Chromatic

MCVP A

Alternating Graph

MFVS A

MinPlus Circuit Value Accessibility Problem A

Restricted Lexicographically

Problem A

First Indep endent Set B

Minimum FeedbackVertex Set A

Restricted Plane Sweep

MinPlusCVP A

Triangulation B

MISH B

RLFMIS B

MM B

SAMCVP A

Mo dInverse B

SBFS A

Mo dPower B

SCH B

Mo dular Inversion B

SC A

Mo dular Powering B

SE A

Monotone Circuit Value

SFI A

Problem A

Short Vectors Dimension B

MP B

Short Vectors B

NAND Circuit Value Problem A

SI A

NANDCVP A

Nearest Neighbor Traveling

SLPmem A

Salesman Heuristic A

SLPnonempty A

NNTSH A

SM B

OHDVR A

Stable Marriage B

OLDVR A

Stack Breadth First Search A

Ordered High Degree Vertex

StraightLine Program

Removal A

Memb ership A

Ordered Low Degree Vertex

StraightLine Program

Removal A

Nonempty A

Ordered Vertices Remaining A

Subgroup Containment A

OVR A

Subgroup Equality A

Path Systems A

Subgroup Finite Index A

PATH A

Successive Convex Hulls B

PCVP A

SV B

Perfect Matching Existence B

SV B

PHCS A

SWEEP B

Planar Circuit Value Problem A

Sylow Subgroups B

Plane Sweep Triangulation A

SylowSub B

PME B

Synchronous Alternating

Prop ositional Horn Clause

Monotone Fanout CVP A

Satisability A

TopCVP A

PST A

Top ologically Ordered Circuit

RCAGAP A

Value Problem A

RCL A

TPFPA A

Real Analogue to CVP A Triviality Problem for Finitely

Presented Algebras A

RealCVP A

Relative Primeness B TVLP B

A Comp endium of Problems Complete for P Preliminary RCS Revision

Two Player Game A

TwoVariable Linear

Programming B

TwoWayDPDA Acceptance A

USATFLIP A

UNSATFLIP A

ULVL B

UMS A

Unication A

Uniform Word Problem for

Finitely Presented Algebras A

Uniform Word Problem for

Lattices A

UNIF A

Unit Length VisibilityLayers B

Unit Resolution A

UNIT A

Univariate Polynomial

Factorization over Q B

Unweighted SATFLIP A

Unweighted MAXCUTSWAP A

Unweighted STABLE NET A

Unweighted SWAP A

Unweighted NotAllEqual

Clauses SATFLIP A

UPFQ B

USN A

USWAP A

UWPFPA A

UWPL A

VisibilityLayers A

VL A