Structure in Approximation Classes

Extended abstract

P Crescenzi V Kann Silvestri and Trevisan

Dipartimento di Scienze dellInformazione

Universitadegli Studi di Roma La Sapienza

Via Salaria Rome Italy

Email fpilucsilvertrevisangdsiuniromait

2

Department of Numerical Analysis and Computing Science

Royal Institute of Technology

S Sto ckholm Sweden

Email viggonadakthse

Summary The study of the approximability prop erties of NPhard optimization problems has

recently made great advances mainly due to the results obtained in the eld of pro of checking

The last imp ortant breakthrough has b een obtained in where the APXcompleteness of

several imp ortant optimization problems is proved thus reconciling two distinct views of ap

proximation classes syntactic and computational In this pap er we obtain new results on the

structure of two imp ortant computationallydened classes the class NPO that is the class of

optimization problems whose underlying decision problem is in NP and the class APX that

is the class of constantfactor approximable NPO problems In particular we give the rst ex

amples of natural APXintermediate problems and the rst examples of natural NPOcomplete

problems Moreover we state new connections b etween the approximability prop erties and the query complexity of NPO problems

Introduction

In his pioneering pap er on the approximation of combinatorial optimization problems

David Johnson formally introduced the notion of approximable problem prop osed approxi

mation for several problems and suggested a p ossible classication of optimization

problems on grounds of their approximability prop erties Since then it was clear that even

though NPhard optimization problems are manyone p olynomialtime reducible to each

other they do not share the same approximability prop erties The main reason of this fact is

that manyone reductions not always preserve the ob jective function and even if this happ ens

they rarely preserve the quality of the solutions It is then clear that a stronger kind of

ducibility has to b e used Indeed an approximation preserving reduction not only has to map

instances of a problem A to instances of a problem B but it also has to b e able to come back

from go o d solutions in B to go o d solutions in A Surprisingly the rst denition of this

kind of reducibility was given as long as years after Johnsons pap er and after that at

least seven dierent denitions of approximation preserving reducibility app eared in the liter

ature see Fig These denitions are identical with resp ect to the overall scheme but dier

essentially in the way they preserve approximability they range from the Strict reducibility

in which the error cannot increase to the PTASreducibility in which there are basically no

restrictions see also Chapter of

PTASreducibility

Preducibility

Areducibility

HY HY

H H

H H

H H

Continuous reducibility Lreducibility Ereducibility

Strict reducibility

Figure The taxonomy of approximation preserving reducibilities

By means of these reducibiliti es several notions of completeness in approximation classes

have b een introduced and basically two dierent approaches were followed On the one hand

the attention was fo cused on computationally dened classes of problems whose approxima

bility prop erties were well understo o d such as NPO and APX along this line of research

however almost all completeness results dealt either with articial optimization problems or

with problems for which lower b ounds on the quality of the approximation were easily obtain

able On the other hand researchers fo cused on the logical denability of optimization

problems and introduced several syntactically dened classes for which natural completeness

results were obtained unfortunately the approximability prop erties of the prob

lems in these latter classes were not related to standard complexitytheoretic conjectures A

rst step towards the reconciling of these two approaches consisted of proving lower b ounds on

the approximability of complete problems for syntactically dened classes unless P NP or

some other unlikely condition More recently another step has b een p erformed since the

closure of syntactically dened classes with resp ect to approximation preserving reducibility

has b een proved to b e equal to the more familiar computationally dened classes

In spite of this imp ortant achievement b eyond APX we are still forced to distinguish b e

tween maximization and minimization problems as long as we are interested in completeness

pro ofs Indeed a result of states that it is not p ossible to rewrite every NP maximization

problem as an NP minimization problem unless NPcoNP A natural question is thus whether

this duality extends to approximation preserving reductions

Finally even though the existence of intermediate articial problems that is problems

for which lower b ounds on their approximation are not obtainable by completeness results

was proved in a natural question arises do natural intermediate problems exist Observe

that this question is also op en in the eld of decision problems even though the existence of

articial NPintermediate problems has b een already proved For example it is known that

the graph isomorphism problem cannot b e NPcomplete unless the p olynomialtime hierarchy

collapses but no similar result has ever b een obtained showing that the problem do es not

b elong to P

Summary of the Results

The rst goal of this pap er is to dene an approximation preserving reducibility such that

all reductions that have app eared in the literature still hold and such that it can b e used

for as many approximation classes as p ossible In spite of the fact that the Lreducibili ty

has b een the most widely used so far we will give strong evidence that it cannot b e used

to obtain completeness results in computationally dened classes such as APX logAPX

that is the class of problems approximable within a logarithmic factor and p olyAPX that

is the class of problems approximable within a p olynomial factor Indeed on the one hand

the Lreducibil ity is to o weak and is not approximation preserving unless P NP coNP

on the other it is to o strict and do es not allow to reduce problems which are known to b e

NP

easy to approximate to problems which are known to b e hard to approximate unless P

NPO log n

P The weakness of the Lreducibil ity is essentially shared by all reducibilitie s of

Fig but the Strict reducibility and the Ereducibili ty while the strictness of the Lreducibili ty

is shared by all of them but the PTASreducibility The reducibility we prop ose is a combination

of the Ereducibility and of the PTASreducibility and as far as we know it is the strictest

reducibility that allows to obtain all approximation completeness results that have app eared

in the literature such as for example the APXcompleteness of the maximum satisability

problem and the p olyAPXcompleteness of the maximum clique problem

The second group of results refers to the existence of natural complete problems in NPO In

deed b oth and provide examples of natural complete problems for the class of minimiza

tion and maximization NP problems resp ectively In Sect we will show the existence of both

maximization and minimization NPOcomplete natural problems In particular we prove that

Maximum Programming Minimum Programming and Minimum Weighted

Independent are NPOcomplete This result shows how making use of a

natural approximation preserving reducibility is enough p owerful to encompass the duality

problem raised in Moreover the same result can also b e obtained when restricting our

selves to the class NPO PB that is the class of p olynomially b ounded NPO problems In

particular we prove that Maximum PB Programming Minimum PB Pro

gramming and Minimum Independent Dominating Set are NPO PBcomplete Indeed

this result can also b e obtained as a consequence of Theorem a of However our proof

does not make use of the PCP model

The third group of results refers to the existence of natural APXintermediate problems

In particular in Sect we will prove that Minimum Bin Packing and other natural NPO

problems cannot b e APXcomplete unless the p olynomialtime hierarchy collapses Since it is

wellknown that this problem b elongs to APX and that it do es not b elong to PTAS that

is the class of NPO problems with p olynomialtime approximation schemes unless PNP our

result thus yields the rst example of a natural APXintermediate problem under a natural

complexitytheoretic conjecture Roughly sp eaking the pro of of our result is structured into

two main steps In the rst step we show that if Minimum Bin Packing is APXcomplete then

the problem of answering any set of k nonadaptive queries to an NPcomplete problem can b e

reduced to the problem of approximating an instance of Minimum Bin Packing within a ratio

dep ending on k In the second step we show that the problem of approximating an instance

of Minimum Bin Packing within a given p erformance ratio can b e solved in p olynomialtime

by means of a constant number of nonadaptive queries to an NPcomplete problem These

two steps will imply the collapse of the query hierarchy which in turn implies the collapse of

the p olynomialtime hierarchy As a side eect of our pro of we will show that if a problem is

APXcomplete then it does not admit an asymptotic approximation scheme as far as we know

no general technique to obtain this kind of results was previously known

In the last group of results we state new connections b etween the approximability prop er

ties and the query complexity of NPhard optimization problems In several recent pap ers the

notion of query complexity that is the number of queries to an NP oracle needed to solve a

given problem has b een shown to b e a very useful to ol for understanding the complexity of

approximation problems In upp er and lower b ounds have b een proved on the number of

queries needed to approximate certain optimization problems such as the maximum satisa

bility problem and the maximum clique problem these results dealt with the complexity of

approximating the value of the optimum solution and not with the complexity of computing

approximate solutions In this pap er instead the complexity of constructive approximation

will b e addressed by considering the languages that can b e recognized by p olynomialtime ma

chines which have a function oracle that solves the approximation problem In particular in

Sect we will b e able to solve an op en question of proving that nding the vertices of the

largest clique is more dicult than merely nding the vertices of a approximate clique that

is a clique with at least half the size of the largest clique unless the p olynomialtime hierarchy

collapses On the one hand the results of show that the query complexity is a go o d

measure of complexity to study approximability prop erties of optimization problems On the

other our results show that completeness in approximation classes implies lower b ounds on the

query complexity In Sect we nally show that the two approaches are basically equivalent by

giving sucient and necessary conditions for approximation completeness in terms of query

complexity hardness and combinatorial properties The imp ortance of these results is twofold

they give new insights into the structure of complete problems in approximation classes and

they reconcile the approach based on standard computation mo dels with the approach based on

the computation mo del for approximation prop osed by As a nal observation our results

can b e seen as an extension of some results of in which general sucient conditions for

APXcompleteness are proved

Due to the lack of space the pro ofs of our results are all contained in the app endix where the problems mentioned in the text are also dened

Preliminaries

Since several introductory b o oks on computational complexity theory make some

mention of approximation classes we will start the following preliminaries on approximation

classes with some mention of computational complexity theory

NP f n

Denition A language L belongs to the class P if it is decidable by a polynomial

time oracle Turing machine which asks at most f n queries to an NPcomplete oracle where

n is the input size

S

NP k

The class QH is equal to the union P Similarly we can dene the function classes

k

NPf n

FP

NPf n NP f n

Theorem For any function f n O log n if P P then the

polynomialtime hierarchy collapses

We now give some standard denitions in the eld of optimization and approximation theory

Denition An NP optimization problem A is a fourtuple I sol m g oal such that

I is the set of the instances of A and it is recognizable in polynomial time

Given an instance x of I sol x denotes the set of feasible solutions of x These solutions

are short that is a polynomial p exists such that for any y sol x jy j pjxj Moreover

it is decidable in polynomial time whether for any x and for any y such that jy j pjxj

y sol x

Given an instance x and a feasible solution y of x mx y denotes the positive integer

measure of y often also called the value of y The function m is computable in polynomial

time and is also called the ob jective function

g oal fmax ming

The class NPO is the set of all NP optimization problems

The goal of an NPO problem with resp ect to an instance x is to nd an optimum solution

that is a feasible solution y such that mx y g oal fmx y y sol xg

In the following opt will denote the function mapping an instance x to the measure of an

optimum solution Max NPO is the set of maximization NPO problems and Min NPO is the

set of minimization NPO problems

An NPO problem is said to b e polynomially bounded if a p olynomial q exists such that for

any instance x and for any solution y of x mx y q jxj The class NPO PB is the set of

all p olynomially b ounded NPO problems NPO PB Max PB Min PB where Max PB is

the set of all maximization problems in NPO PB and Min PB is the set of all minimization

problems in NPO PB

Denition Let A be an NPO problem Given an instance x and a feasible solution y of x

we dene the p erformance ratio of y with resp ect to x as

mx y optx

Rx y max

optx mx y

The p erformance ratio is always a number greater than and is as close to as the feasible solution is close to the optimum one

Denition Let A be an NPO problem and let T be an that for any instance x

of A returns a feasible solution T x Given an arbitrary function r N we say that

T is an r napproximate algorithm for A if for any instance x the performance ratio of the

feasible solution T x with respect to x veries the fol lowing inequality

Rx T x r jxj

Denition Given a class of functions F an NPO problem A belongs to the class F APX

if an r napproximate polynomialtime algorithm T for A exists for some function r F

In particular APX logAPX and p olyAPX will denote the classes F APX with F equal to

the set of constant functions to the set O log n and to the set of p olynomials resp ectively

Denition An NPO problem A belongs to the class PTAS if an algorithm T exists such

that for any xed rational r T r is a polynomialtime r approximate algorithm for A

Clearly the following inclusions hold

PTAS APX logAPX p olyAPX NPO

It is also easy to see that these inclusions are strict if and only if P NP

A new approximation preserving reducibility

We will justify our denition by emphasizing the disadvantages of previously known reducibil

ities

The Lreducibility

The rst reducibility we shall consider is the Lreducibili ty for linear reducibility which is

often most practical to use in order to show that a problem is at least as hard to approximate

as another

Denition Let A and B be two NPO problems A is said to be Lreducible to B in symbols

A B if two functions f and g and two positive constants and exist such that

L

For any x I f x I is computable in polynomial time

A B

For any x I and for any y sol f x g x y sol x is computable in polynomial

A B A

time

For any x I opt f x opt x

A B A

For any x I and for any y sol f x

A B

jopt x m x g x y j jopt f x m f x y j

A A B B

Clearly the Lreducibili ty preserves membership in PTAS However the next result gives a

strong evidence that in general this reducibility is not approximation preserving It also shows

that the b ehavior of Lreductions dep ends on the type that is maximization or minimization

of the problems involved

Theorem The fol lowing hold

Lreductions from minimization problems to optimization problems are approximation pre

serving

Lreductions from maximization problems to optimization problems are not approximation

preserving if and only if the reducibility is dierent from the manyone reducibility

Observe that in it is shown that the hypothesis of the p oint ab ove is somewhat inter

mediate b etween P NP coNP and P NP In other words there is strong evidence that

even though the Lreducibil ity is suitable to prove APXcompleteness results this reducibility

cannot b e used to dene the notion of completeness within classes b eyond APX Moreover it

cannot b e used to obtain p ositive results that is the existence of approximation algorithms

via reductions

The Ereducibility

The drawbacks of the Lreducibili ty are mainly due to the fact the relation b etween the p er

formance ratios not necessarily linear is obtained by putting separate linear constraints on

the relations b etween b oth the optimum values and the absolute errors The Ereducibili ty for

error reducibili ty instead imp oses a linear relation directly b etween the p erformance ratios

Denition Let A and B be two NPO problems A is said to be Ereducible to B in symbols

A B if two functions f and g and a positive constant exist such that

E

For any x I f x I is computable in polynomial time

A B

For any x I and for any y sol f x g x y sol x is computable in polynomial

A B A

time

For any x I and for any y sol f x

A B

R x g x y R f x y

A B

Observe that for any function r an Ereduction maps r napproximate solutions into

r n approximate solutions so that it not only preserves membership in PTAS but also

membership in any F APX class where F is closed with resp ect to linear applications such

as p olyAPX logAPX and APX As a consequence of this observation and of the results of

the previous section we have that NPO problems should exist which are Lreducible to each

other but not Ereducible However the following result shows that within the class APX the

Ereducibili ty is just a generalization of the Lreducibil ity

Prop osition For any two NPO problems A and B if A B and A APX then A B

L E

Clearly the converse of the ab ove result do es not hold since no problem in NPO NPO PB

can b e Lreduced to a problem in NPO PB while any problem in PO can b e Ereduced to any

NPO problem

The Ereduction is still somewhat to o strict Indeed the next result shows that unless

NP NPO log n

P P PTAS problems exist which are not reducible to APX problems observe

that from the ab ove prop osition this fact holds for the Lreducibil ity as well Intuitively this

unnatural b ehavior is due to the fact that an Ereduction preserves optimum values

Prop osition Maximum Knapsack is not Ereducible to any NPO PB problem unless

NP NPO log n

P P

The APreducibility

We have just observed that a drawback of the Ereducibili ty consists of preserving optimum

solutions This is due to the fact that the linear relation b etween the p erformance ratios is

to o restrictive According to the denition of approximation preserving reducibiliti es given in

we could overcome this problem by expressing this relation by means of an implication

However this solution is not sucient intuitively since the function g do es not know which

approximation is required it must still map optimum solutions into optimum solutions The

nal step thus consists in letting the function g dep end on the p erformance ratio Indeed in

the following denition which is a restriction of the PTASreducibility introduced in we

also let the function f dep end on this ratio b ecause this feature will turn out to b e useful in

order to prove interesting characterizations of complete problems in approximation classes

Denition Let A and B be two NPO problems A is said to be APreducible to B in

symbols A B if two functions f and g and a positive constant exist such that

AP

For any x I and for any r f x r I

A B

For any x I for any r and for any y sol f x r g x y r sol x

A B A

f and g are computable by two algorithms T and T respectively whose running time is

f g

polynomial for any xed r

For any x I for any r and for any y sol f x r

A B

R f x r y r implies R x g x y r r

B A

Observe that clearly the APreducibili ty is a generalization of the Ereducibility Moreover

it is easy to see that Prop osition do es not hold for the APreducibili ty indeed any PTAS

problem is APreducible to any NPO problem As far as we know this reducibili ty is the

strictest one app earing in the literature that allows to obtain natural APXcompleteness results

for instance the APXcompleteness of Maximum Satisfiability

NPOcomplete problems

We will in this section prove that there are natural problems that are complete for the classes

NPO and NPO PB Previously completeness results have b een obtained just for Max NPO

Min NPO Max PB and Min PB One example of such a result is the following

theorem

Theorem Minimum Weighted Satisfiability is Min NPOcomplete and Max

imum Weighted Satisfiability is Max NPOcomplete even if only a subset fv v g of

s

si

the variables has nonzero weight and w v for i s

i

We will construct APreductions from maximization problems to minimization problems

and vice versa Using these reductions we will show that a problem that is Max NPOcomplete

or Min NPOcomplete in fact is complete for the whole of NPO and that a problem that is

Max PBcomplete or Min PBcomplete is complete for the whole of NPO PB

Theorem Minimum Weighted Satisfiability and Maximum Weighted Satisfiabil

ityare NPOcomplete

Corollary Any Min NPOcomplete problem is NPOcomplete and any Max NPOcomplete

problem is NPOcomplete The problems Minimum Programming Traveling Sales

person Problem and Minimum Weighted Independent Dominating Set are NPO

complete

We can also show that there are natural complete problems for the class of p olynomially

b ounded NPO problems

Theorem Maximum PB Programming and Minimum PB Programming

are NPO PBcomplete

Corollary Any Min PBcomplete problem is NPO PBcomplete and any Max PBcomplete

problem is NPO PBcomplete

By using the construction of the pro of of Theorem together with the result that Longest

Induced Path is not approximable within jV j for any unless P NP one can

show the following new hardness results for some NPO PBcomplete problems

Theorem The fol lowing problems are not approximable within n for any unless

P NP Maximum Number of Satisfiable Formulas n is the number of equations

Maximum Distinguished Ones n is the number of distinguished variables Maximum

PB Programming n is the number of inequalities

The fol lowing problems are not approximable within n for any unless P NP

Maximum PB Programming n is the number of variables Maximum Constrained

Binary Satisfiable Linear Subsystem n is the number of variables

The fol lowing problems are not approximable within n for any unless P NP Max

imum Ones n is the number of variables Maximum Irrelevant Binary Variables

in Linear System n is the number of variables

Query complexity and APXintermediate problems

Denition Let A be an NPO problem and r be a function then A is the fol lowing

r n

multivalued partial function given an instance x of A A x is the set of feasible solutions

r n

y of x such that Rx y r jxj

A

r

Denition Given an NPO problem A and a rational r a language L belongs to P

if two polynomialtime computable functions f and g exist such that for any x f x is an

instance of A and for any y A f x g x y if and only if x L

r

S

A

r

Using techniques similar to those of P The class AQHA is equal to the union

r

we can prove the following result

Prop osition For any problem A in APX AQHA QH

Recall that an NPO problem admits an asymptotic polynomialtime approximation scheme

if an algorithm T exists such that for any x and for any r Rx T x r r k optx

with k constant and the of T x r is p olynomial with resp ect to jxj The

class of problems that admit an asymptotic p olynomialtime approximation scheme is usually

denoted as PTAS The following result shows that for this class the previous fact can b e

strengthened

NPh

Then a constant h exists such that AQHA P Prop osition Let A PTAS

The following fact instead states that any language L in the query hierarchy can b e decided

using just one query to A where A is APXcomplete and dep ends on the level of the query

hierarchy L b elongs to

Prop osition For any APXcomplete problem A QH AQHA

By combining Prop ositions and we thus have the following result that characterizes the

approximation query hierarchy of the hardest problems in APX

Theorem For any APXcomplete problem A AQHA QH

Finally as a consequence of this theorem of Prop osition of Theorem and of the results

of we have the following result

Corollary If the polynomialtime hierarchy does not collapse then Minimum Degree

Spanning Tree Minimum Bin Packing and Minimum Edge Coloring are APX

intermediate

A remark on Maximum Clique

The following two prop ositions are the analogous of Prop ositions within NPO

A NPO log n

r

Prop osition For any NPO problem A and for any r P P

NPlog log nO A

r

P Prop osition For any NPO PB problem A and for any r P

NPlog n MC

1

From Prop osition from the fact that P is contained in P where MC stands

for Maximum Clique and from Theorem it thus follows the next result that solves an

op en question p osed in Informally this result states that it is not p ossible to reduce the

problem of nding a maximum clique to the problem of nding a approximate clique unless

the p olynomialtime hierarchy collapses

MC MC

1 2

Theorem If P P then the polynomialtime hierarchy collapses where MC stands

for Maximum Clique

Query complexity and completeness in approximation classes

In this nal section we shall give a full characterization of problems complete for p olyAPX

and for APX resp ectively in terms of query complexity

NP q n

Denition NPF is the class of partial multivalued functions computable by non

deterministic polynomialtime Turing machines which ask at most q n queries to an NP oracle

in the entire computation tree

Denition Let F and G be two partial multivalued functions We say that F manyone

reduces to G in symbols F G if two polynomialtime algorithms t and t exist such

mv

that for any x in the domain of F t x is in the domain of G and for any y Gt x

t x y F x

1

We say that a multivalued partial function F is computable by a nondeterministic Turing machine N if

for any x in the domain of F an halting computation path of N x exists and any halting computation path of

N x outputs a value of F x

NP q n NP q n

We shall say that a function F is hard for NPF if for any G NPF G F

mv

The following denition is a constructive version of the denition of selfimprovability given

in

Denition A problem A is selfimprovable if two algorithms t and t exist such that for

any instance x of A and for any two rational r r x t x r r is an instance of A

and for any y A x y t x y r r A x Moreover for any xed r and r the

r r

2 1

running time of t and t is polynomial

From it follows that the equivalence with resp ect to the APreducibili ty preserves the

selfimprovability prop erty We are now ready to state the main result of this section

Theorem A polyAPX problem A is polyAPXcomplete if and only if it is selfimprovable

NP log log nO

is NPF hard for some r and A

r

0

The ab ove theorem cannot b e proved without the dep endency of b oth f and g on r in the

denition of APreducibili ty Indeed it is p ossible to prove that if only g has this prop erty

then unless the p olynomialtime hierarchy collapses a selfimprovable problem A exists such

NPlog log nO

that A is NPF hard and A is not p olyAPXcomplete

In order to characterize APXcomplete problems we have to dene the following combina

torial prop erty

Denition An NPO problem A is linearly additive if a constant and two algorithms t

and t exist such that for any rational and for any sequence x x of instances of A

k

x t x x is an instance of A and for any y A x t x x y

k k

k

y y where each y is a approximate solution of x Moreover the running time of

k i i

t and t is polynomial for every xed

Theorem An APX problem A is APXcomplete if and only if it is linearly additive and

NP

is NPF hard a constant r exists such that A

r

0

Note that linear additivity plays for APX problem more or less the same role of self

improvability in p olyAPX These two prop erties are in a certain sense one the opp osite of the

other while the query complexity of APXcomplete problems dep ends on the p erformance ratio

and do es not dep end on the size of the instance the query complexity of p olyAPXcomplete

problems dep ends on the size of the instance and do es not dep end on the p erformance ratio

Indeed it is p ossible to prove that no APXcomplete problem can b e selfimprovable un

less P NP and that no p olyAPXcomplete problem can b e linearly additive unless the

p olynomialtime hierarchy collapses

It is also p ossible to establish query complexity results for logAPXcomplete problem In

particular even though we have not b een able to establish a full characterization of logAPX

complete problems we can prove the following result

Theorem No logAPXcomplete problem can be selfimprovable unless the polynomial

timehierarchy collapses

It is then an interesting op en question to nd a characterizing combinatorial prop erty of log

APXcomplete problems Moreover as a consequence of the ab ove theorem and of the results of we conjecture that the minimum set cover problem is not selfimprovable

References

Amaldi E and Kann V The complexity and approximability of nding maximum feasible

subsystems of linear relations Technical Rep ort ORWP Department of Mathematics Swiss

Federal Institute of Technology Lausanne Theoretical Comput Sci to app ear

Amaldi E and Kann V On the approximability of removing the smallest number of

relations from linear systems to achieve feasibility Technical Rep ort ORWP Department of

Mathematics Swiss Federal Institute of Technology Lausanne

Arora S Lund C Motwani R Sudan M and Szegedy M Pro of verication and

hardness of approximation problems Proc of rd Ann IEEE Symp on Foundations of Comput

Sci IEEE Computer So ciety

Berman P and Schnitger G On the complexity of approximating the indep endent set

problem Inform and Comput

Bovet DP and Crescenzi P Introduction to the theory of complexity Prentice Hall

Chang R On the query complexity of clique size and maximum satisability Proc th

Ann Structure in Complexity Theory Conf IEEE Computer So ciety

Chang R A machine mo del for NPapproximation problems and the revenge of the

Bo olean hierarchy EATCS Bul letin

Chang R Gasarch WI and Lund C On b ounded queries and approximation Tech

nical Rep ort TR CS Department of Computer Science University of Maryland Baltimore

County

Crescenzi P and Panconesi A Completeness in approximation classes Inform and

Comput

Crescenzi P and Trevisan L On approximation scheme preserving reducibility and its

applications Proc th FSTTCS Lecture Notes in Comput Sci SpringerVerlag

F urer M and Raghavachari B Approximating the minimum degree spanning tree to

within one from the optimal degree Proc Third Ann ACMSIAM Symp on Discrete Algorithms

ACMSIAM

Garey MR and Johnson DS Computers and intractability a guide to the theory of

NPcompleteness Freeman

Holyer I The NPcompleteness of edgecoloring SIAM J Computing

Impagliazzo R and Naor M Decision trees and downward closures Proc rd Structure

in Complexity Theory Conference IEEE Computer So ciety

Johnson DS Approximation algorithms for combinatorial problems J Comput System

Sci

Kann V On the approximability of NPcomplete optimization problems PhD thesis De

partment of Numerical Analysis and Computing Science Royal Institute of Technology Sto ckholm

Kann V Polynomially b ounded minimization problems that are hard to approximate

Nordic J Computing

Karmarkar N and Karp R M An ecient approximation scheme for the onedimensional

Proc of rd Ann IEEE Symp on Foundations of Comput Sci IEEE

Computer So ciety

Khanna S Motwani R Sudan M and Vazirani U On syntactic versus computational

views of approximability Proc of th Ann IEEE Symp on Foundations of Comput Sci IEEE

Computer So ciety

Kolaitis P G and Thakur M N Approximation prop erties of NP minimization classes

Proc Sixth Ann Structure in Complexity Theory Conf IEEE Computer So ciety

Krentel MW The complexity of optimization problems J Comput System Sci

Ladner RE On the structure of p olynomialtime reducibility J ACM

Lund C and Yannakakis M On the hardness of approximating minimization problems

J ACM

Lund C and Yannakakis M The approximation of maximum subgraph problems

Proc of th International Col loquium on Automata Languages and Programming Lecture Notes

in Comput Sci SpringerVerlag

Motwani R Lecture notes on approximation algorithms Technical Rep ort STANCS

Department of Computer Science Stanford University

Orp onen P and Mannila H On approximation preserving reductions Complete problems

and robust measures Technical Rep ort C Department of Computer Science University

of Helsinki

Panconesi A and Ranjan D Quantiers and approximation Theoretical Computer

Science

Papadimitriou CH Computational complexity AddisonWesley

Papadimitriou C H and Yannakakis M Optimization approximation and complexity

classes J Comput System Sci

Schoning U Graph isomorphism is in the low hierarchy Proc th Ann Symp on

Theoretical Aspects of Comput Sci Lecture Notes in Comput Sci SpringerVerlag

Simon HU Continuous reductions among combinatorial optimization problems Acta

Informatica

Valiant L Relative complexity of checking and evaluating Information Processing

Letters

Wagner K Bounded query computations Proc rd Ann Structure in Complexity

Theory Conf IEEE Computer So ciety

App endix

We will now give the pro ofs of the results presented in the pap er the denitions of the

problems and some additional references

A Pro of of the results of Section

Proof of Theorem follows from the fact noted in that if a minimization problem

A Lreduces to an optimization problem B and there is a p olynomialtime r approximation

algorithm for B then there is a p olynomialtime r for

A

In order to prove rst recall that in it has b een shown that the reducibility

is dierent from the manyone reducibility if and only if a p olynomialtime recognizable set

of satisable Bo olean formulas exists for which no p olynomialtime algorithm can compute

a satisfying assignment for each of them Assume now that there exists a p olynomialtime

recognizable set S of satisable Bo olean formulas for which no p olynomialtime algorithm

can compute a satisfying assignment for each of them Consider the following maximization

problems A I sol m and B I sol m where

A A A B B B

I S sol x fy jy is a truth assignment for xg and

A A

jxj if y is a satisfying assignment for x

m x y

A

otherwise

and

I S sol x fy jy is a truth assignment for xg and

B B

jxj if y is a satisfying assignment for x

m x y

B

jxj otherwise

Clearly problem B is in APX while if A were in APX then there were a p olynomialtime

algorithm that computes a satisfying assignment for each formula in S contradicting the as

sumption Moreover it is easy to see that A is Lreducible to B

Conversely assume that for any p olynomialtime recognizable set of satisable Bo olean

formulas there is a p olynomialtime algorithm computing a satisfying assignment for each

formula in the set

Supp ose that a maximization problem A is Lreducible to a maximization problem B via

functions f and g and that B is r approximable r Let x b e an instance of A and let y

b e a solution of f x such that opt f xm f x y r For the sake of convenience set

B B

opt opt x m m x g x y opt opt f x and m m f x y We show

A A A A B B B B

that m maxfm m g is such that m opt and opt m r that is m is

A B A A

r

a nonconstructive approximation of opt Let There are two cases

A

r

opt opt By the denition of the Lreducibili ty opt m opt m Since

B A A A B B

opt m opt m

B B A A

r Hence opt opt and opt m r it holds that

B A B B

opt opt

A B

opt m opt m r

A A A

opt opt It holds that

B A

opt opt

A A

m m

B

opt

B

since opt opt

A B

m

B

opt

B

since m opt r

B B

opt r

B

r

r

Now it is not hard to see that a satisable Bo olean formula can b e constructed in p olynomial

time in the length of x so that any satisfying assignment for enco des a solution of x whose

measure is at least m By assumption it is p ossible to compute in p olynomial time a satisfying

assignment for and thus an approximate solution for x

Finally if B is a minimization problem we rst Lreduce B to a maximization problem C

in APX and then apply the ab ove argument ut

Proof of Proposition Let T b e an r approximation algorithm for A with r constant and

let f g b e an Lreduction from A to B Then for any x I and for any y

L L L L A

sol f x E x g x y E f x y where

B L A L L L B L

joptw mw z j

E w z

optw

denotes the relative error of the feasible solution z with resp ect to the instance w If A is a

minimization problem then for any x I and for any y sol f x

A B L

R x g x y E x g x y E f x y R f x y

A L A L L L B L L L B L

Otherwise we distinguish the following two cases

E f x y in this case we have that

B L

L L

E x g x y

A L

R x g x y

A L

E x g x y

A L

E f x y

L L B L

E f x y

L L B L

R f x y

L L B L

in this case we have that R f x y so that E f x y

B L B L

L L L L

R x T x r r R f x y

A L L B L

We can thus dene the Ereduction f g as follows

E E E

For any x I f x f x

A E L

For any x I and for any y sol f x

A B E

g x y if m f x y m f x T x

L B E B E

g x y

E

T x otherwise

maxf r g

E L L L L

From the ab ove discussion it follows that this reduction is indeed an Ereduction ut

NP MK

1

Proof of Proposition From the results of it follows that P P where MK stands

for Maximum Knapsack If Maximum Knapsack is Ereducible to an NPO PB problem

MK A A NPO log n

1 1 1

A then P P It is easy to see that P P which in turn implies that

NP NPO log n

P P ut

B Pro of of the results of Section

Proof of Theorem In order to establish the NPOcompleteness of Minimum Weighted

Satisfiability we just have to show that there is an APreduction from a Max NPOcomplete

problem to Minimum Weighted Satisfiability As the Max NPOcomplete problem we will

use the restricted version of Maximum Weighted Satisfiability from Theorem

Let x b e an instance of Maximum Weighted Satisfiability ie a formula over vari

si

ables v v with weights w v and some variables with weight zero We will rst

s i

give a simple reduction that preserves the approximability within the factor and then adjust

it to obtain an APreduction

Let f x b e the formula where z v v v where

s i i i i

i

z z are new variables with weights w z for i s and where all other variables

s i

even the v variables have zero weight If y is a satisfying assignment of f x let g x y b e the

restriction of the assignment to the variables that o ccur in This assignment clearly satises

Note that exactly one of the z variables is true in any satisfying assignment of f x If all

z variables were false then all v variables would b e false and the value of the ob jective function

of x would b e zero which is not allowed

i

mf x y z

i

v v v v

i i

si si

mx g x y

s s

mx g x y

mf x y mf x y

This is in particular true for the optimum solution Thus the p erformance ratio for Maximum

Weighted Satisfiability is

s

mf x y optx

optf x

Rf x y Rx g x y

s

mx g x y optf x

mf x y

which means that the reduction preserves the approximability within

k

Let us now extend the construction in order to obtain Rx g x y Rf x y

k

for every nonnegative integer k The reduction describ ed ab ove corresp onds to k

k

We use s new variables named z where i s and b f g for j k

ib b j

1

k

V

Let f x where

k ib b

1

k

z v v v v b v b

ib b ib b i i i ik k

1 1

k k

Dene g x y as ab ove Finally dene

s i

K K

w z

P P

ib b

1

k

k k

j

b w v b w v

j ij j i

j j

s

we can disregard the eect of the ceiling op eration in By choosing K large enough ab out

the following computations

As in the previous reduction exactly one of the z variables is true in any satisfying assign

then we have mf x y w z ment of f x If in a solution y of f x z

k ib b k k ib b

1 1

k k

and we know that

k k

X X

j si

b b w v mx g x y w v

j j ij i

j j

and that

k s k

X X X

j k si

b w v b w v mx g x y w v

j j j ij i

j j

j k i

Thus we get

s s

K K

k

mx g x y

mf x y mf x y

k k

k

and therefore Rx g x y Rf x y Given any r if we choose k such that

k

k

r r eg k dlog r log r e then Rf x y r implies Rx g x y

k

k k

Rf x y r r r r r This is obviously an APreduction

k

with

A very similar pro of can b e used to show that Maximum Weighted Satisfiability is

NPOcomplete ut

Proof of Corollary Theorem says that Minimum Weighted Satisfiability is NPO

complete Per denition Minimum Weighted Satisfiability can b e reduced to any Min NPO

complete problem Hence any Min NPOcomplete problem is also complete for NPO In the

same way since Maximum Weighted Satisfiability is NPOcomplete and can b e reduced

to any Max NPOcomplete problem any Max NPOcomplete problem is NPOcomplete

Min NPOcompleteness for Minimum Programming and Traveling Salesperson

Problem was shown in and therefore they are NPOcomplete For Minimum Weighted

Independent Dominating Set we need a PTASreduction from Minimum Weighted Sat

isfiability

Given an instance of Minimum Weighted Satisfiability we rst write the formula in

a with weights For every variable v we construct two no des a and

i i i

w a w a w v For every clause c we construct a no de b with enormous weight We

i i i j j

P

s

w v choose a weight larger that the sum of the weights of all variables namely

i

i

a for each i For each i and j we also add edges as follows If We add edges b etween a and

i i

the variable v is used p ositively in clause c we add an edge b etween the corresp onding no des

i j

a and b If the variable v is used negatively in clause c we add an edge b etween a and b

i j i j i j

One solution to the indep endent dominating set problem is fa a g and this solution

s

has smaller ob jective value than any solution that contains a bno de It is easy to see that in

a for any i is included that the corresp onding assignment any solution at most one of a and

i i

v if and only if a is included satises the CNF formula and that the ob jective values

i i

of b oth solutions are the same The reduction is PTASpreserving ut

Proof of Theorem Maximum PB Programming is known to b e Max PBcomplete

and Minimum PB Programming is known to b e Min PBcomplete Thus we just

have to show that there are APreductions from a Min PBcomplete problem to Maximum PB

Programming and from a Max PBcomplete problem to Minimum PB Program

ming As the Min PBcomplete problem we will use Minimum Independent Dominating

Set and as the Max PBcomplete problem we will use Longest Induced Path

Both reductions follow the same idea The ob jective function ie the number of no des in the

solution in the indep endent dominating set and in the induced path resp ectively is enco ded

by introducing an order of the no des in the solution The order is enco ded by a squared number

of variables in the programming problem see Fig A solution of size shall corresp ond

to the programming ob jective value n and a solution of size p shall corresp ond to an

j k

n

ob jective value of

p

j

only zeros in upp er part

size of

one in each row

solution

i

solution

Figure The idea of the reduction from Minimum Independent Dominating SetLongest In

j

if and only if v duced Path to MaximumMinimum PB Programming The variable x

i

i

is the j th no de in the solution There is at most one in each column and in each row

The reduction from Minimum Independent Dominating Set to Maximum PB

Programming is constructed as follows Given an instance of Minimum Independent

Dominating Set ie a graph with no des V fv v g and edges E construct m

m

j

variables x i j m and the following inequalities

i

m

P

j

i m x at most one in each column

i

j

m

P

j

x at most one in each row j m

i

i

m m

P P

j j

x only zeros in upp er part x j m

i i

i i

m m

P P

k k

x indep endence x v v E

i j

j i

k k

m

P P

k k

x domination x i m

j i

k

j v v E

i j

k m

The ob jective function is dened as

m m

X X

n n

p

n x

i

p p

p i

In order to express the ob jective function with only binary co ecients we have to introduce n

P

p

m

x for bnpc j bnp c and y for new variables y y where y

j n j

i

i

P

n

j bnmc The ob jective function is then y One can now verify that an indep endent

j

j

dominating set of size s will exactly corresp ond to a solution of the programming problem

n

and vice versa with ob jective value

s

Supp ose that the minimum indep endent dominating set has size M then the p erformance

ratio sM for the indep endent dominating set problem will corresp ond to the p erformance

ratio

n

m s

M

n

M n

s

m

is the relative error due to the o or op eration for the programming problem where

n

By choosing n large enough the relative error can b e made arbitrarily small Thus It is easy to

see that the reduction is an APreduction

Halldorsson has proved that unless P NP Minimum Independent Dominating Set

is not approximable within n for any where n is the sum of the number of no des

and edges in the graph Together with the reduction ab ove this result will tell that unless

P NP Maximum PB Programming is not approximable within q for any

where q is the number of inequalities and is not approximable within r for any

where r is the number of variables

A similar construction can b e used to reduce Longest Induced Path to Minimum PB

Programming Given an instance of Minimum Independent Dominating Set ie

j

a graph with no des V fv v g and edges E construct m variables x i j m as

m

i

ab ove and use the inequalities together with the new inequalities

l k

induced x v v E k m l k m x

i j

j i

k

k

v v E k m x x path

i j

i

j

Use the same ob jective function as ab ove It is not hard to show that this reduction is a

APreduction ut

The results of Theorem are proved in

C Pro of of the results of Section

Proof of Proposition Assume that A is a maximization problem the pro of for minimization

problems is similar let T b e an r approximate p olynomialtime algorithm for A for some

A

for some Two p olynomialtime computable functions f and g then r and let L P

exist witnessing this fact For any x let m mf x T f x so that m optf x r m

We can then partition the interval m r m into dlog r e subintervals

blog r c

m m m m r m

and start lo oking for the subinterval containing the optimum value a similar technique has

b een used in This can clearly b e done using dlog r e queries to an NPcomplete oracle

One more query is sucient to know whether a feasible solution y exists in that interval such

that g x y Since y is approximate it follows that L can b e decided using dlog r e

queries that is L QH ut

Proof of Proposition Let A b e a minimization problem in PTAS the pro of for maximization

problem is very similar By denition a constant k and an algorithm T exist such that for

any instance x and for any r

mx T x r r optx k

NPh

We will now prove that a constant h exists such that for any r a function l PF

r

exists such that for any instance x of the problem A

optx l x r optx

r

Given an instance x we can check whether optx by means of a single query to an

NP oracle so we can restrict ourselves to instances such that optx Note that for these

instances T is a k approximate algorithm Let us x an r let r y T x

and m mx T x We have to distinguish two cases

app

m k k in this case optx k that is optx Then

app

mx y optx optx optx r optx

That is y is a r approximate solution for x and we can set l x mx y

r

mx y k k in this case optx k k Then

mx y optx optx k optx k k k

If h dlog k k e then h queries to NP are sucient to nd the optimum value

m optx by means of a binary search technique in this case l x m

r

A

r

for some r let f and g b e the Let now L b e a language in AQH A then L P

A

r

functions witnessing that L P Observe that for any x x L if and only if a solution y

for f x exists such that mx y l x and g x y that is given h x deciding whether

r r

x L is an NP problem Since l x is computable by means of h queries to NP we have

r

h

that L P ut

In order to prove Prop osition we need the following technical result

Claim For any APXcomplete problem A and for any k two p olynomialtime computable

functions f and g and a constant r exist such that for any k tuple x x of instances of

k

Partition x f x x is an instance of A and if y is a solution of x whose p erformance

k

ratio is smaller than r then g x y b b where b f g and b if and only if x

k i i i is a yesinstance

Proof Let x U s b e an instance of Partition for i k Without loss of generality

i i i

P

we can assume that the U s are pairwise disjoint and that for any i s u Let w

i i

uU

i

b e an instance of Minimum Ordered Bin Packing dened as follows a similar construction

has b een used in in order to prove negative results on the approximability of Minimum

Ordered Bin Packing

S

k

U fu u g where the u s are new items U

i k i

i

For any u U su s u and su for i k

i i i

For any i j k for any u U and for any u U u u u

i j i

Any solution of w must b e formed by a sequence of packings of U U such that for

k

any i the bins used for U are separated by the bins used for U by means of one bin which

i i

is completely lled by u In particular the packings of the U s in any optimum solution must

i i

use either two or three bins two bins are used if and only if x is a yesinstance The optimum

i

measure thus is upp er b ounded by k so that any k approximate solution is

optimum

Since Minimum Ordered Bin Packing b elongs to APX and A is APXcomplete then

an APreducibili ty f g exists from Minimum Ordered Bin Packing to A We can

then dene x f x x f w k and r k For any r approximate

k

solution y of x the fourth prop erty of the APreducibili ty implies that z g x y k

is a k approximate solution of w and thus an optimum solution of w From z we can

easily derive the right answers to the k queries x x ut

k

NP h

Proof of Proposition Let L QH then L P It is well known that L can b e reduced to

h

the problem of answering k nonadaptive queries to NP More formally two functions

t and t exist such that for any x t x x x where x x are k instances of the

k k

k

Partition problem and for any b b f g t x b b f g Moreover if

k k

for any j b if and only if I Partition then t x b b if and only if x L

j j k

Let now f g and r b e the two functions and the constant from the preceding Claim applied

to problem A and constant k For any x x f t x is an instance of A such that if y is a

A

r

r approximate solution for x then t g x y if and only if x L Thus L P ut

Prop ositions and can b e easily proved similarly to Prop osition by means of the binary

search technique

D Pro of of the results of Section

Proof of Theorem Let A b e a p olyAPXcomplete problem Since Maximum Clique is

selfimprovable and p olyAPXcomplete we have that A is selfimprovable It is then

NPlog log nO

sucient to prove that A is hard for NPF

Since A is p olyAPXcomplete Maximum Clique A let b e the constant of

AP

NPlog log nO

this reduction From we have that any function F in NPF manyone re

duces to Maximum Clique From the denition of APreducibili ty we also have that

Maximum Clique A so that F manyone reduces to A

mv

Conversely let A b e a p olyAPX selfimprovable problem such that for some r A

r

0

NPlog log nO

is NPF hard We will show that for any problem B in p olyAPX B is

APreducible to A To this aim we introduce the following partial multivalued function

multisat given in input a sequence of instances of the satisability problem

m



with m log j j a p ossible output is a satisfying truthassignment for where

m i

NPlog log nO

i maxfij is satisable g From it follows that this function is NPF

i

complete

B B

It is easy to see that since B is in p olyAPX two algorithms t and t exist such that

B B

for any xed r t x t x r and t x t x r are a manyone reduction from

NPlog log nO

B to multisat Moreover since A is NPF hard then a manyone reduction

r r

0

M M A A

t t exists from multisat to A Finally let t and t b e the functions witnessing the

r

0

selfimprovability of A

The APreduction from B to A can then b e derived as follows

B M 0 A 00

t xr t x t x r r

0

1 1 1

x r x x x

0 B 0 00 M 00 000 A

xy r t x y t x y r r t

0

2 2 2

y y y y

It is easy to see that if y is an r approximate solution for the instance x of A then y is an

r approximate solution of the instance x of B ut

Recall that in an extension of Theorem is proved that is for any q n O log n if

NPq n NP q n

NPF is contained in NPF then the p olynomial hierarchy collapses

NPq n

In the same pap er a characterization of the classes NPF is given such that any

NPk

k

function F NPF is reducible to answering with witnesses a set of nonadaptive

queries to NP Moreover it is easy to see that any function that is reducible to answering

NPk

k

with witnesses nonadaptive queries to NP is contained in NPF From the pro ofs of

Prop ositions and it follows the following fact

Claim Let A b e an r approximable APX problem then for any r A is reducible to

A r

answering with witnesses a set of dlog r e nonadaptive queries to an NP oracle Let A b e an

A

r

APXcomplete problem then a constant exists such that for any k the problem of answering

with witnesses a set of k nonadaptive queries to Partition is reducible to A

k

Proof of Theorem Let A b e an r approximable APXcomplete problem let b e the

A

NP

constant in the preceding Claim then A is hard for NPF Fix any r let r

and let x x b e instances of A for any i k the problem of nding a r approximate

k

solution y for x is reducible to the problem of answering with witnesses a set of dlog r e

i i A

r

parallel queries to Partition Without loss of generality we can assume r r otherwise

A

the reduction is trivial and thus we have dlog r e r r c for a

A A

r

certain constant Moreover answering k c nonadaptive queries to Partition is reducible

to k capproximating a single instance of A that is A is linearly additive

NP

Conversely let A b e a linearly additive APX problem such that A is NPF hard and

r

0

let B b e an r approximable APX problem Given an instance x of B for any r

B

we can reduce the problem of nding an approximate solution for x to the problem of

answering with witnesses c queries to Partition for a prop er constant c not dep ending on

NP

Moreover each of these questions is reducible to A since an NPF can clearly answer with

r

0

witness to an NP query From linear additivity it follows that r approximating c instance

of A is reducible to capproximating a single instance of A This is an APreduction

from B to A with c ut

Proof of Theorem Let us consider the optimization problem Max Number of Satisfiable

Formulaslog dened as follows

Instance Set of m b o olean formulas in CNF such that is a tautology and

m

m log j j

m

Solution Truthvalue assignment to the variables of

m

Measure The number of satised formulas ie jfi such that is satised by gj

i

Clearly Max Number of Satisfiable Formulaslog is in logAPX since the measure of

any assignment is at least and the optimum value is always smaller that log n where n is

NPlog log log n

the size of the input We will show that for any r MNSF is hard for NPF

r

where MNSF stands for Max Number of Satisfiable Formulaslog

Given log log n queries to NP of size p olynomial in n we can construct

log log n

an instance of Max Number of Satisfiable Formulaslog where is a

m

tautology and the formulas i i+1 are satisable if and only if at least i clauses

among these formulas can b e easily constructed using the standard pro of of

log log n

log log n

Co oks theorem Note that m and by adding dummy clauses to some formulas

we can achieve the b ound m log j j Moreover from a rapproximate solution for

m

we can decide how many clauses in are satisable and we can also recover

log log n

NP log log log n

witnesses for such formulas that is any function in NPF is MVreducible to

MNSF

r

Let A b e a selfimprovable logAPXcomplete problem then for any function F

NPlog log log n

NPF F MNSF A A where is the constant in the AP

mv mv mv

reduction from Max Number of Satisfiable Formulaslog to A Thus for any x instance

of F computing F x is reducible to nding a approximate solution for an instance x of

c

A moreover the size of x is p olynomial in jxj that is jx j jxj for a certain constant c Since

A l og APX it is p ossible to nd in p olynomial time a r log jx japproximate solution y

for x By means of the usual binary search technique we can nd a approximate solution

r c

for x using dlog dlog r log jx jee log log log jxj adaptive queries to NP Thus

r c

NPlog log log n NP log log log jxj

NPF NPF

which implies the collapse of the p olynomial hierarchy ut

E A List of NPO Problems

E Maximum Clique

Instance Graph G V E

Solution A clique in G ie a subset V V such that every two vertices in V are joined

by an edge in E

Measure Cardinality of the clique ie jV j

E Minimum Independent Dominating Set

Instance Graph G V E

Solution An indep endent dominating set for G ie a subset V V such that for all

u V V there is a v V for which u v E and such that no two vertices in V are

joined by an edge in E

Measure Cardinality of the indep endent dominating set ie jV j

E Minimum Weighted Independent Dominating Set

Instance Graph G V E and a weight function w V N

Solution An indep endent dominating set for G ie a subset V V such that for all

u V V there is a v V for which u v E and such that no two vertices in V are

joined by an edge in E

P

Measure The sum of the weights of the indep endent dominating set ie w v

0

v jV j

E Maximum Weighted Satisfiability and Minimum Weighted Satisfiability

Instance Set of variables X b o olean quantierfree rstorder formula over the variables

X and a weight function w X N

Solution Truth assignment that satises

Measure The sum of the weight of the satised variables

E Maximum Programming and Minimum Programming

mn m

Instance Integer m nmatrix A Z integer mvector b Z nonnegative integer

n

nvector c N

n

Solution A binary nvector x f g such that Ax b

n

X

Measure The scalar pro duct of c and x ie c x

i i

i

E Maximum PB Programming and Minimum PB Programming

mn m

Instance Integer m nmatrix A Z integer mvector b Z nonnegative binary

n

nvector c f g

n

Solution A binary nvector x f g such that Ax b

n

X

Measure The scalar pro duct of c and x ie c x

i i

i

F A List of APX Problems

F Minimum Bin Packing

Instance Finite set U of items and a size su Q for each u U

Solution A partition of U into disjoint sets U U U such that the sum of the sizes of

m

the items in each U is at most

i

Measure The number of used bins ie the number of disjoint sets m

F Minimum Ordered Bin Packing

Instance Finite set U of items a size su Q for each u U and a partial order

on U

Solution A partition of U into disjoint sets U U U such that the sum of the sizes of

m

the items in each U is at most and for any u U and for any u U such that u u

i i j

i j

Measure The number of used bins ie the number of disjoint sets m

F Maximum Knapsack

Instance Finite set U for each U a size su Z and a value v u Z a p ositive integer

B Z

P

su B Solution A subset U U such that

0

uU

P

Measure Total weight of the chosen elements ie v u

0

uU

G Additional references

HalldorssonM M Approximating the minimum maximal indep endence number Inform

Process Lett

Kann V Strong lower b ounds of the approximability of some NPO PBcomplete max

imization problems Technical Rep ort TRITANA Department of Numerical Analysis and

Computing Science Royal Institute of Technology Sto ckholm

Long TJ On reducibility versus p olynomial time manyone reducibility Theoretical

Computer Science

Queyranne M Bounds for assembly line balancing heuristics Operations Research