The Complexity of Approximating PSPACE Complete Problems
; ;
for Hierarchical Sp eci cations
Madhav V Marathe Harry B Hunt III S S Ravi
Department of Computer Science
University at Albany State University of New York
Albany NY
Abstract
We extend the concept of p olynomial time approximation algorithms to apply to problems for hier
archically sp eci ed graphs many of which are PSPACE complete Assuming P 6 PSPACE the exis
tence or nonexistence of such e cient approximation algorithms is characterized for several standard
graph theoretic and combinatorial problems We present p olynomial time approximation algorithms
for several standard PSPACE hard problems considered in the literature In contrast we show that
unless P PSPACE there is no p olynomial time approximation for any for several other
problems when the instances are sp eci ed hierarchicall y
We present p olynomial time approximation algorithms for the following problems when the graphs
are sp eci ed hierarchical ly
minimum vertex cover maximum SAT weighted max cut minimum maximal matching and
bounded degree maximum independent set
In contrast we show that unless P PSPACE there is no p olynomial time approximation for
any for the following problems when the instances are sp eci ed hierarchicall y
the number of true gates in a monotone acyclic circuit when al l input values are speci ed and
the optimal value of the objective function of a linear program
It is also shown that unless P PSPACE a p erformance guarantee of less than cannot b e obtained
in p olynomial time for the following problems when the instances are sp eci ed hierarchicall y
high degree subgraph k vertex connected subgraph and k edge connected subgraph
Classi cation Hierarchical Sp eci cations Approximation Algorithms Computational Complexity Al
gorithms and Data structures
1
Email addresses of authors fmadhav hunt ravig cs albany edu
2
Supp orted by NSF Grants CCR and CCR
3
An extended abstract of this pap er app eared in the pro ceedings of th International Col loquium on Automata Languages
and Programming ICALP
Intro duction
Hierarchical system design is b ecoming increasingly imp ortant with the development of VLSI technology
HLW RH At present a numb er of VLSI circuits already have over a million transistors For
example the Intel i chip has ab out million transistors Although VLSI circuits can have millions
of transistors they usually have highly regular structures These regular structures often make them
amenable to hierarchical design sp eci cation and analysis Other applications of hierarchical sp eci cation
and consequently of hierarchically sp eci ed graphs are in the areas of nite element analysis LW b
software engineering GJM material requirement planning and manufacturing resource planning in a
multistage pro duction system MTM and pro cessing hierarchical Datalog queries Ul
Over the last decade several theoretical mo dels have b een put forward to succinctly represent ob jects
hierarchically BOW GW Le Le LW Wa Here we use the mo del de ned by Lengauer in
HLW Le Le LW to describ e graphs Using this mo del Lengauer et al Le LW a Le
have given e cient algorithms to solve several graph theoretic problems including minimum spanning
forests planarity testing etc
Here we extend the concept of p olynomial time approximation algorithms so as to apply to problems
for hierarchically sp eci ed graphs including PSPACE complete such problems We characterize the exis
tence or nonexistence assuming P PSPACE of p olynomial time approximation algorithms for several
standard graph problems Both p ositive and negative results are obtained see Tables and at the end
of this section Our study of approximation algorithms for hierarchically sp eci ed problems is motivated
by the following two facts
(n)
n size hierarchical sp eci cations can sp ecify size graphs
Many basic graph theoretic prop erties are PSPACE complete HR LW rather than NP
complete
For these reasons the known approximation algorithms in the literature are not directly applicable to
graph problems when graphs are sp eci ed hierarchically
What we mean by a polynomial time approximation algorithm for a graph problem when the graph is
sp eci ed hierarchically can b e b est understo o d by means of an example
Example Consider the minimum vertex cover problem where the input is a hierarchical sp eci cation
of a graph G We provide e cient algorithms for the following versions of the problem
The Approximation Problem Compute the size of a near minimum vertex cover of G
The Query problem Given any vertex v of G and the path from the ro ot to the no de in the
hierarchy tree see Section for the de nition of hierarchy tree in which v o ccurs determine whether
v b elongs to the approximate vertex cover so computed
The Construction Problem Output a hierarchical sp eci cation of the set of vertices in the
approximate vertex cover
The Output Problem Output the approximate vertex cover computed
Our algorithms for and ab ove run in time p olynomial in the size of the hierarchical
sp eci cation rather than the size of the graph obtained by expanding the sp eci cation Our algorithm
for runs in time linear in the size of the expanded graph but uses space which is linear in the size of
the hierarchical sp eci cation
This is a natural extension of the de nition of approximation algorithms in the at i e non hierarchical
case This can b e seen as follows In the at case the numb er of vertices is p olynomial in the size of the de
scription Given this any p olynomial time algorithm to determine if a vertex v of G is in the approximate
minimum vertex cover can b e mo di ed easily into a p olynomial time algorithm that lists all the vertices
of G in the approximate minimum vertex cover For an optimization problem or a query problem our
algorithms use space and time which are low level p olynomials in the size of the hierarchical sp eci cation
and thus O pol y log in the size of the sp eci ed graph when the size of the graph is exp onential in
the size of the sp eci cation Moreover when we need to output the subset of vertices subset of edges
etc corresp onding to a vertex cover maximal matching etc in the expanded graph our algorithms take
essentially the same time but substantially less often exp onentially less space than algorithms that work
directly on the expanded graph It is imp ortant to design algorithms which work directly on the hierar
chical sp eci cation by exploiting the regular structure of the underlying graphs b ecause graphs resulting
from expansions of given hierarchical descriptions are frequently to o large to t into the main memory of
a computer Le This results in a large numb er of page faults while executing the known algorithms on
the expanded graph Hence standard algorithms designed for at graphs are impractical for hierarchically
sp eci ed graphs
We b elieve that this is the rst time e cient approximation algorithms with go o d p erformance guaran
4
tees have b een provided b oth for hierarchically sp eci ed problems and for PSPACE complete problems
Thus by providing algorithms which exploit the underlying structure we extend the range of applicability
of standard algorithms so as to apply to a much larger set of instances Tables and summarize our
results
4
Indep endently Condon et al CF a CF a have investigated the approximability of other PSPACE complete
problems
Table Performance Guarantees
Problem Performance guarantee Best known guarantee
in hierarchical case in at case
MAX SAT
MIN Vertex
Cover
MIN Maximal
Matching
Bounded Degree B B B
MAX Indep endent Set
MAX CUT
The results mentioned in the last column of the ab ove table can b e found in GJ Ya
Table Hardness Results
Problem Hierarchical Flat Case
Case
Maximum Numb er PSPACE hard Log hard for
of True Gates to approximate P to approximate
in a circuit for any for any
Optimal Value of PSPACE hard Log hard for
Ob jective Function for P to approximate
of a Linear Program any for any
High Degree PSPACE hard Log hard for
Subgraph for P to approximate
for
k Vertex PSPACE hard Log hard for
Connectivity for P to approximate
for
k Edge PSPACE hard Log hard for
Connectivity for P to approximate
for
The results mentioned in the last column of the ab ove table can b e found in AM KSS Se
De nitions and Description of the Mo del
The following two de nitions are from Lengauer Le
De nition A hierarchical speci cation G G of a graph is a sequence of undirected simple
1 n
graphs G cal led cel ls The graph G has m edges and n vertices p of the vertices are distinguished
i i i i i
and are cal led pins The other n p vertices are cal led inner vertices r of the inner vertices are
i i i
distinguished and are cal led nonterminals The n r vertices are cal led terminals
i i
Note that there are n p r vertices de ned explicitly in G We call these explicit vertices Each pin
i i i i
of G has a unique lab el its name The pins are assumed to b e numb ered from to p Each nonterminal in
i i
G has two lab els a name and a type The typ e is a symb ol from G G If a nonterminal vertex v is
i 1 i 1
of the typ e G then the terminal vertices which are the neighb ors of G are in one to one corresp ondence
j j
with the pins of G Note that all the neighb ors of a nonterminal vertex must b e terminals Also a
j
terminal vertex may b e a neighb or of several nonterminal vertices The size of denoted by siz e is
P P
m n and the edge numb er M N M where the vertex numb er N
i i
1in 1in
De nition Let G G be a hierarchical speci cation of a graph G The expansion E
1 n
i e the graph associated with of the hierarchical speci cation is done as fol lows
k E G
1
k Repeat the fol lowing step for each nonterminal v of G say of the type G delete v and the edges
k j
th
incident on v Insert a copy of E by identifying the l pin of E with the node in G that is labeled
j j k
v l The inserted copy of E is cal led the subcel l of G Observe that the expanded graph can have
j k
multiple edges although none of the G have multiple edges
i
The expansion E is the graph asso ciated with the hierarchical de nition Note that the total
(N )
numb er of no des in E can b e For i n G G is the hierarchical sp eci cation of
i 1 i
the graph E Given a hierarchical sp eci cation one can asso ciate a natural tree structure depicting
i
the sequence of calls made by the successive levels We call it the hierarchy tree and denote it by HT
A vertex in E is identi ed by a sequence of nonterminals on the path from the ro ot to the nonterminal
in which the vertex is explicitly de ned For the query problems considered in the pap er we assume that
a vertex is sp eci ed in the ab ove manner
Without loss of generality we assume that there are no useless cells in
n
Example Figure shows an example of a hierarchically sp eci ed graph and its corresp onding hierarchy
tree The lab els on the vertices are omitted and the corresp ondence b etween the pins of G and the
j
neighb ors of a nonterminal of typ e G in the cell G is clear by the p ositions of the vertices in the gure
j i
Figure shows the underlying graph E G We note again that our approximation algorithms answer
query problems without explicitly expanding the hierarchical speci cation
De nition A hierarchical graph speci cation G G of a graph G is level restricted
1 n
if for al l u v E one of the fol lowing conditions holds
u and v are explicit vertices in the same instance of G i n
i
u is an explicit vertex in an instance of G and v is a explicit vertex in an instance of G and the
i j
instance of G cal ls the instance of G j i n
i j
A hierarchical graph sp eci cation G G of a graph G is strongly level restricted if it
1 n
is level restricted and in addition for i n the only nonterminals of G are of the typ e G
i i 1 α γ 1 5
G1 β G1
2 3 4
G G 2 1
G 3 G1 a c d
b G G2 1 G1 G G1 2 e
G1 G 1
G3 HierarchyTree HT( G 3 )
explicit vertices pins non-terminals
Figure A hierarchically sp eci ed graph and the corresp onding hierarchy tree
a b d
c e 1 5
2 3 4 γ
α β
1 5 1 5 1 5
2 3 4 2
2 3 4 3 4
Figure The graph asso ciated with the hierarchical sp eci cation G
The ab ove de nition can b e extended to de ne k level restricted sp eci cations for any xed k Such
descriptions still can lead to exp onentially large graphs Moreover many practically o ccurring hierarchical
descriptions see Le Le LW a are k level restricted for small values of k We note that our
PSPACE hardness results hold for strongly level restricted sp eci cations while all our approximation
algorithms hold for arbitrary sp eci cations
De nition Let G G be a hierarchical speci cation is said to be simple if for each G
1 n i
i n there are no edges between pins de ned in G
i
For a simple level restricted sp eci cations observe that
Observation Consider any edge u v in a simple level restricted hierarchical speci cation of a
graph G Then the path from u to v in the hierarchy tree passes through at most one pin
For the rest of the discussion given a problem we denote by the same problem when the instance
HG
is sp eci ed hierarchically So for example we use M AX C U T to denote the MAX CUT problem when
HG
the graph is sp eci ed hierarchically Also we sometimes use the phrase hierarchical graphs to mean
hierarchically sp eci ed graphs
Finally we give additional de nitions used in the pap er
De nition The Monotone Circuit Value Problem MVCP is de ned as fol lows Given an acyclic
graph G cal led the circuit with one distinguished vertex output the sources inputs labeled with f g
and al l other vertices labeled with symbols from f g the decision version of the problem asks if the
output of G is The optimization version of MCVP denoted by MTG asks for the maximum number of
gates which are set to
We assume that the reader is familiar with the problem SAT The problem SAT is de ned as
HG
follows
1 n 1 n
De nition An instance F F X F X F X of SAT is of the form
1 n 1 n HG
i i i i i
F X X Z f X Z F
i i i
j
j j
1j l
i
n i i i i
for i n where f are CNF formulae X X X Z Z i n are vectors of boolean
i
j j
i i i i
variables such that X X Z Z i i Thus F is just a CNF formula An instance of
j 1
j j
SAT speci es a CNF formula f that is obtained by expanding the F j n as macros where
HG j
the variables Z s introduced in any expansion are considered distinct The problem SAT is to decide
HG
whether the formula f speci ed by F is satis able The optimization problem denoted by MAX SAT
HG
is to nd an assignment to the variables of f satisfying the maximum number of clauses in f
i i
Let n b e the total numb er of variables used in F i e jX j jZ j and let m b e the total numb er of
i i i
P
m n clauses in F The size of F denoted by siz e F is equal to
i i i
1in
Example Let F F x x F x x F b e an instance of SAT where each F is de ned as
1 1 2 2 3 4 3 HG i
follows
F x x x x z z z
1 1 2 1 2 1 2 3
F x x F x z F z z z z x
2 3 4 1 3 4 1 4 5 4 5 4
F F z z F z z
3 2 8 7 1 7 6
1 1 1 2 2 2 3 3
The formula f denoted by F is z z z z z z z z z z z z z z
7 6 8 4 4 5
1 2 3 1 2 3 1 2
3
z z z z
4 5 7
3
De nition Let F be an instance of the problem SAT with set of variables V and set of clauses C
The bipartite graph of F denoted BG f is the bipartite graph V C E where e c v E i
variable v occurs in clause c
F is said to be planar i the graph BG f is planar
1 n 1 n
De nition An instance F F X F X F X of Hierarchical
1 n 1 n 1 n 1 n
Linear Program LP is of the form
HG
i i i i i
F X X Z f X Z F
i i i
j
j j
1i i
j
X X
c z d
j j i i i
j j
i
i
z 2Z
j
j
n i i i
for i n where f is a set of linear inequalities X X X Z Z i n are vectors
i j
j j
i i i i
of variables such that X X Z Z i i F is a set of linear inequalities and is a linear
j i i
j j
objective function over the variables in E F Thus F is just a set of linear inequalities An instance
i 1
of LP de nes a hierarchical ly speci ed linear program F obtained after expanding F j n as
HG n j
macros where the Z s in di erent expansions are considered distinct and a linear objective function
n
0
obtained after expanding s as macros
j
Let n b e the total numb er of variables used in F and let m b e the total numb er of inequalities
i i i i
P
m n in F Then the size of F denoted by siz e F is equal to
i i i
1in
The LP feasibility problem is to determine whether there exists an assignment to the variables over
the reals used in the LP such that all the inequalities are satis ed In the case of the LP optimization
HG
problem one is given a linear ob jective function and linear inequalities b oth de ned hierarchically as
ab ove The aim is to nd an assignment to the variables so as to maximize the value of the ob jective
function sub ject to the inequality constraints Using Lengauer s de nition of hierarchical graphs one can
represent a LP graphically by asso ciating a no de with each variable and with each inequality Further
HG
a variable no de has an edge to an inequality no de i the corresp onding variable o ccurs in the inequality
Linear programming has b een extensively studied in literature In GLS it is shown how linear
programs can b e used to mo del many graph theoretic problems In GLS it was also shown that for the
class of perfect graphs p olynomial time algorithms can b e devised to compute an optimal vertex coloring
maximum indep endent set and several other imp ortant graph theoretic parameters When graphs are
represented hierarchically the corresp onding linear program will b e hierarchical But as will b e shown
Section computing the optimal value of the ob jective function of a hierarchically sp eci ed linear
program is PSPACE hard further it is also PSPACE hard to compute an approximate value of the
ob jective function
Next we recall the de nitions of high degree subgraph and high vertex edge connectivity problems
De nition The High Degree Subgraph Problem k HDSP is de ned as fol lows For al l
integers k given a graph G V E does G have a nonempty subgraph of minimum degree k The
optimization problem of k HDSP denoted by MAX HDSP asks for the maximum k such that there is a
vertex induced subgraph of G in which the minimum degree of a vertex is k
Let HDSP denote the largest k such that there is an induced subgraph of minimum degree k An
approximate solution to this problem is a subgraph in which each no de has degree at least d where HDSP
d HDSP c for some xed c For all k k HDSP was shown to b e log complete for P in
AM Furthermore unless P NC it was shown that no NC approximation algorithm for MAX
HDSP could provide a p erformance guarantee b etter than k HDSP is p olynomial time solvable for at
graphs AM We show that k HDSP is PSPACE complete and furthermore unless P PSPACE
HG
MAX HDSP cannot b e approximated with a factor c in p olynomial time See Section The high
HG
degree subgraph problem contrasts with the related maximum clique problem MCP which is NP complete
for b oth at GJ and hierarchically sp eci ed graphs LW
Next we recall the de nitions the high vertex and edge connectivity problems from KSS
De nition The vertex connectivity G edge connectivity G of an undirected graph G is the
5
minimum number of vertices edges whose removal results in a disconnected or a trivial graph A graph
is m vertex connected m edge connected if G m G m
De nition The High Vertex Connectivity Problem HVCP High Edge Connectivity
Problem HECP is de ned as fol lows For al l integers given a graph G V E does
G contain an induced subgraph of vertex connectivity edge connectivity at least The optimization
5
A trivial graph consists solely of isolated vertices
versions of these problems denoted by MAX HVCP MAX HECP ask for the largest for such that there
is an induced subgraph of vertex edge connectivity
Let HVCP HECP denote the largest such that there is an induced subgraph of vertex edge
connectivity An approximate solution to this problem is a subgraph whose vertex edge connectivity
is at least d where HVCP HECP d HVCP c HECP c for some xed c It was shown in
KSS that for all HVCP and HECP are log complete for P Furthermore they showed that
Theorem Kirousis Serna Spirakis KSS Unless P NC MAX HVCP and MAX HECP cannot
be approximated to within a factor c of the optimal in NC
Here we show that for all the problems HVCP and HECP are PSPACE complete and
HG HG
furthermore unless P PSPACE MAX HVCP and MAX HECP cannot b e approximated within a
HG HG
factor c in p olynomial time See Section
We end this section with a few comments regarding our approximation algorithms for the problems
MAX CUT MAX SAT and Bounded degree Indep endent set when instances are sp eci ed hierarchically
Consider the MAX CUT problem For any graph G V E there is always a cut containing at least jE j
edges Therefore by merely counting the numb er of edges in a hierarchically sp eci ed graph one can
always compute a numb er which is within a factor of of an optimal cut However our approximation
algorithm for the MAX CUT problem actual ly nds a hierarchical representation of a cut containing at
least jE j edges Similar comments apply to our approximation algorithms for the problems MAX SAT
and Bounded degree Indep endent set when instances are sp eci ed hierarchically
Approximation Algorithms
In this section we discuss our approximation algorithms for the problems given in Table We rst outline
the basic technique used to e ciently obtain approximation algorithms with go o d p erformance guarantee
The Basic Technique Approximate Burning
Our approximation algorithms are based on a new technique which we call approximate burning This
is an extension of the Bottom Up metho d for pro cessing hierarchical graphs discussed in LW a Le
Le and Wi for designing e cient algorithms for hierarchically sp eci ed graphs The b ottom up
b
metho d aims at nding a small graph G called the burnt graph which can replace each o ccurrence of G
i
i
b
in such a way that G and G b ehave identically with resp ect to the problem under consideration The
i
i
b ottom up metho d should pro duce such burnt graphs e ciently Since the problems we are dealing with
are PSPACE hard we cannot hop e to nd in p olynomial time such burnt graphs which can replace original
graphs Therefore we resort to approximate burning In approximate burning given an approximation
algorithm for non hierarchical instances of the problem we wish to nd small burnt graphs which can
b e used to replace the original non terminals in such a way that the p erformance guarantee provided by
the algorithm is not a ected by the replacement All our approximation algorithms rely on approximate
burning
In summary to obtain go o d solutions for a problem sp eci ed hierarchically the b ottom up pro cedure
should have the following prop erties
Each burnt graph should have a size which is p olynomial in the size of the sp eci cation
The burning pro cedure should run in time which is p olynomial in the size of sp eci cation
The burnt graphs should b e replaceable with resp ect to the problem and the approximation
algorithm A
Before we discuss our approximation algorithm we give a transformation which allows us to transform
a hierarchical sp eci cation in which there are edges b etween pins de ned in G to an equivalent hierarchical
i
sp eci cation which has no edges b etween pins de ned in a given G The transformation is outlined in
i
Figure
The following lemma summarizes the prop erty of the sp eci cation obtained as a result of the
1
transformation outlined in Figure
Lemma Given a hierarchical speci cation G G in which there are edges between pins
1 n
de ned in a given G we can construct in polynomial time n new hierarchical speci cation H H
i 1 1 n
such that
siz e is polynomial in siz e
1
can be constructed in polynomial time
1
E E
1
For each H i n there are no edges between pins de ned in H
i i
In view of Lemma we assume that in the input to all our approximation algorithms is a simple
hierarchical sp eci cation i e there is no edge b etween two pins which are de ned in the same cell The
running times of our approximation algorithms are with resp ect to such simple sp eci cation
Approximation Algorithm for Vertex Cover
We now discuss our heuristic for computing the size of a near optimal vertex cover for a hierarchically
sp eci ed graph The problem of computing the size of a minimum vertex cover for hierarchically sp eci ed
graphs was shown to b e PSPACE hard by Lengauer LW Actually they prove the hardness for maxi
mum indep endent set the hardness of minimum vertex cover is therefore directly implied Our heuristic
Pro cedure Transform HSPEC
Input A hierarchical speci cation G G of a graph G
1 n
Output A new hierarchical speci cation H H which has no edges b etween pins de ned in a
1 1 n
given H
i
Phase
a i Initially the graph H is identical to G
1 1
b b
ii The burnt graph G for G is constructed as follows The pins in G are the same as the pins
1
1 1
b
in the original graph There is an edge b etween two pins in G i there is an edge b etween the
1
corresp onding pins in G
1
b Rep eat the following steps for i n
i Let A denote the set of all the terminals pins and explicit vertices in G Let the non
i i
Substitute the burnt graphs for each of the non terminals G terminals called by G b e G
i i i
k 1
0
called in G to obtain G The cell H is obtained as follows The terminals in H are in one
i i i
i
to one corresp ondence with the terminals A in G Furthermore there is an edge b etween two
i i
0
terminals i either there was an edge b etween the corresp onding terminals in the graph G
i
called in G corresp onding to the non terminals G H H also calls non terminals H
i i i i i
k 1 k 1
and the terminal H G The one one corresp ondence b etween the pins of non terminals H
i i i
k 1
r k vertices of H is the same as the one one corresp ondence for G except that for G
i i i
r
r k we substitute H
i
r
b b
ii Construct the burnt graph G as follows The pins in G are the same as the pins in G As in the
i
i i
b b
case of G there is an edge b etween two pins in G i there is an edge b etween the corresp onding
1
i
0
pins in G
i
Phase Mo dify each H i n by removing any edges b etween pins in the de nition of H
i i
Output H H as the new sp eci cation for G
1 1 n
Figure Algorithm for Pro ducing Simple Sp eci cations
builds on the well known vertex cover heuristic for the at non hierarchical case where one computes a
maximal matching and returns all the vertices involved in the matching as an approximate vertex cover
The algorithm in the non hierarchical case has a p erformance guarantee of GJ
We note that the straightforward greedy approach for obtaining a maximal matching in a at graph
cannot b e directly extended to the hierarchical case Two reasons for this are as follows First the degree
of a vertex in a hierarchical graph can b e exp onential in the size of the description and so it is not p ossible
to keep track of the neighb ors of a no de explicitly Secondly an edge b etween a pair of no des can pass
through several pins and thus need not b e explicitly present at any level Therefore edges cannot b e
handled as simply as in the at case This complicates our heuristic since we can keep track of only a
p olynomial amount of information at each level
Before we present the heuristic we give some notation which we use throughout this section Given
a graph G MM G denotes a maximal matching in the subgraph induced by the explicit vertices in G
i e no pins and no nonterminals V MM G denotes the vertices in the subgraph induced by MM G
M xM G denotes a maximum matching of G and V M xM G denotes the vertices in the subgraph
induced by M xM G We use G to denote the size of an approximate vertex cover for E G i e
i i
expanded version of G We also use EM G to denote the set of edges implicitly chosen by the heuristic
i i
from E G
i
The following lemma recalls known prop erties of a maximum matching in a bipartite graph
Lemma Let G S T E be a bipartite graph and let M xM G denote a maximum matching for G
S T S T
Let V and V denote the set of vertices in S and T included in V M xM G Let V and V denote
1 1 2 2
the set of vertices in S and T not included in V M xM G Then the fol lowing statements hold
S T
For al l V and V E
2 2
S T S T
For al l v V v V v V and v V if v v M xM G and v v E then
x y z w x y y z
1 1 2 2
v v E
x w
Pro of
If E then f g M xM G is also a feasible matching This contradicts the assumption
that M xM G is a maximum matching for G
Supp ose v v M xM G v v E and v v E Then the matching M xM G
x y y z x w
f v v g f v v v v g contains more edges than M xM G violating the assumption that
x y y z x w
M xM G is a maximum matching
Figure gives the details of our approximation algorithm for minimum vertex cover
Heuristic HVC
Input A simple hierarchical speci cation G G of a graph G
1 n
Output The size and a hierarchical description of an approximate vertex cover for G
Rep eat the following steps for i n
a Compute MM G
i
Remark Recall that MM G is a maximal matching on the subgraph of G induced on the
i i
explicit vertices in G
i
l l
b Compute V where V denotes the explicit vertices in G not in V MM G Also let G call
i i i
i i
in its de nition Recall that i i j k G non terminals if any G
j i i
k 1
l
are the endp oints of those G which are connected to pins in G Remark Vertices in V
i i
k 1
i
where i i edges that have their other endp oints in one of G
j i
j
l
c For each vertex v V do
i
l
If v is not adjacent to any nonterminals in G then delete v from V else
i
i
b
is called in G such that G G Let v b e adjacent to p
i i i
r r
i
r
r k then match v with x such i If there exists a marked edge incident on any of the p
v i
r
l b
is a marked edge and delete v from V and x from this copy of G that x p
v v i
r
i i
r
else
l b
Delete v from V and y from this copy is an edge in G ii Cho ose a vertex y such that y p
v v v i
r
i i
r
b
of G
i
r
d Let
i l
V fw j w V and w is matched in step c g
x i
i b
and w is matched in step c g V fw j w V G
y i
j
l
e Construct a maximum matching on the set of vertices remaining in V and the burnt graphs of
i
nonterminals called in G
i
b 1 b
G f For the bipartite graph G induced by the vertices left over in G including those in G
i
i i i
k 1
1 b 1
and the pins in G construct M xM G G for G is the vertex induced subgraph of M xM G
i i
i i i
1 b
The edges in M xM G are marked in G
i i
k
X
i i
G g G jV MM G j jV j jV j
i i i
j x y
j =1
i i
Remark Let CM f u v ju V v V and u and v get matched up in Step c g
i
x y
k
Note that EM G is only needed in the pro of it is EM G EM G MM G CM
i i i i i
j
j =1
not explicitly computed Further G jEM G j
i i
i i
h Construct H as follows The explicit vertices in H are the vertices in the set V MM G V V
i i i
x y
i i
Their names are the same as those of the vertices in the sets V MM G V V If G calls a
i i
x y
non terminal G j i then H calls a copy of H
j i j
Remark The H created has the following prop erty
i
Given a vertex v E G as a path in the hierarchy tree it is easy to check if v o ccurs in E H
i i
by simply following the same path It is clear that if v is in the approximate vertex cover then it
will o ccur in a non terminal on the path from the ro ot to the non terminal in which v is de ned
Output G and the hierarchical sp eci cation H H H
n 1 n
Figure Details of Vertex Cover Heuristic
Pro of Of Correctness and Performance Guarantee
We now show that the ab ove algorithm implicitly computes a maximal matching for E G
n
Lemma EM G is a valid matching
n
Pro of We need to show that every vertex u is in at most one edge in EM G
n
Case Vertex u is matched with a vertex v such that b oth u and v are explicitly de ned in G for some
i
i i n This implies that in Step b the edge u v was chosen as an memb er of MM G In
i
Step c we do not consider any vertices which were in V MM G Hence u is not an endp oint of any
i
other edge in EM G
n
Case Vertex u is matched with a vertex v such that u G and v G Without loss of generality
j i
b
assume that j i In this case u was a part of the burnt graph G and G calls G By Step c no edge
i j
j
incident on u has b een chosen in MM G Once u v is chosen then in Step c we do not consider the
i
vertices u and v anymore
Lemma The matching EM G is maximal
n
Pro of We need to prove that each edge in the expanded graph E has at least one of its endp oints in
EM G The pro of consists of an exhaustive case analysis Consider an edge e E T There are two
n
cases
Case Both endp oints of e are explicit vertices in the de nition of a cell G
i
The pro of for this case follows directly from Step of the heuristic and the de nition of MM G
i
Case Let v v denote the edge e such that v is in G and v is in G where j i This edge e
i j i i j j
r p where the path in the hierarchy tree from G to G passes through a sequence of pins p
i i i
r r
see Figure By the de nition of hierarchical sp eci cation it is clear that for G G consists of G
i j i
1 p
each pin in a nonterminal G called in G we have exactly one terminal in G which is adjacent to the
k t t
pin We have two sub cases to consider
Case v V MM G or v V MM G Here the pro of follows from the de nition of maximal
i i j j
matching
Case v V MM G and v V MM G We have two sub cases again
i i j j
b
Case v V M xM G
j
j
In this case we know that v was matched with one of the pins We have to consider two sub cases
j
dep ending on whether the vertex v was a part of the burnt graph for all the non terminal no des on the
j
path from G to G in the hierarchy tree or it was a part of burnt graphs for some non terminal and
j i
subsequently got dropp ed
b
Informally this means that the vertex v was a Case m such that m p v V G
j j
i
m
part of the burnt graph for every non terminal which is on the path from G to G
i j
In this case when we pro cess the cell G either v or v get matched up in Step c Hence the edge v v
i i j i j
is covered
b b
Informally v was not part and v V G Case m m p such that v V G
j j j
i i
m m 1
is on the path from G to G in the hierarchy tree and G of the burnt graph for cell G
i j i i
m m G i
G i p
G i m
G i 1
G j
Figure Figure showing the p osition of G and G in the hierarchy tree
i j
In this case if v gets matched with some other vertex we are done So assume that v is dropp ed i e
j j
v is not a part of the burnt graph Now we need to show that v gets a matching partner when it is
j i
picked up for pro cessing This case is more complicated and the pro of uses the following lemmas which
in turn are proven using Lemma
i i i
m m m b
i e v was not Lemma Let v be adjacent to pins p p and let v V G p in G
j j j i
m
i
i i i
m
1 2 k
picked up as a matching partner for any of the pins Then the fol lowing statements hold
i i
m m
Each pin p is matched with a distinct vertex v l k
i i
l l
i
i i
m
m m
is not adjacent to any pin p such that p l l k v does not have a matching partner in
r r
i
l
b
G
i
m
Pro of
Follows from the fact that we computed a maximum matching in Step e and of Lemma
Follows from of Lemma
i i i i
m m m m
if v Call a vertex v is matched up with p a private partner of a pin p and is not in G
i
m
i i i i
l l l l
i
m
which do es not have a matching partner The following lemma says that in G adjacent to any pin p
i
m r
each of the subsequent pins which are on the path from v to v has a if v gets dropp ed o at stage G
j i j i
m
private matching partner
i b b
m
which is adjacent to v and be a pin in G Let p and v V G Lemma Let v V G
j i j j
m x
i i
m m 1
i
i
i p i
m+1
i
m m
m
p terminates at v Then each of the pins p p p to v has a on the path from p
i i
x
1 2 1 p m+1
b
m q p private partner in G
i
q
Pro of By induction on the length of the path from G to G in the hierarchy tree HT
i j i
i
m
has a private partner Basis The path is of length By and of Lemma it follows that p
1
Induction Assume that the Lemma holds for all paths of length Now consider a path of length
i
m
Again by Lemma p is matched up with say v By and of Lemma we know that v is
k k
1
i
m
We therefore have only two cases to consider the private partner of p
1
i
m+1
Case v gets matched up with p
k
2
In this case we can use our induction hyp othesis and we are done
Case v gets dropp ed
k
i
m+1
By and of Lemma we know that the pin p will get some other private partner Now by
2
Induction hyp othesis we are done 2
We now continue the pro of of Case By Lemma it follows that when G is pro cessed pin
i
i
p
p has a private partner Therefore when we pro cess v v is sure to get matched up b ecause G
i i i
p
p m+1
i
p
the private partner of p which is adjacent to v cannot b e used as matching partner by any other
i
p m+1
vertex in G So that the edge v v is covered by the vertex v
i i j i
b
Case v V M xM G The argument is similar to that of Case b ecause v gets dropp ed
j j
j
at the very rst stage
Theorem Given a hierarchical graph G the above approximation algorithm computes an approximate
vertex cover within factor of of the optimal value
Pro of Follows from Lemmas and
Query Problem
We can easily mo dify our algorithm to answer the query problem For this we can use the hierarchical
representation of the solution obtained
Theorem Given any vertex v as a path in the hierarchy tree we can determine in O N M if v is
in the approximate vertex cover so computed
Pro of Observe that the hierarchy tree for H is identical to the hierarchy tree for except that the no des
in HT H are lab eled by H whenever the corresp onding no de in HT is lab eled G This means that
i i
the sequence of nonterminals used to identify the query vertex v can b e used to to check if v is in the
approximate vertex cover computed For this note that if v is in the approximate vertex cover then it
lies on the path from the ro ot of HT H to a nonterminal H such that v is in the corresp onding G in
i i
the original graph G
The hierarchical sp eci cation can b e used to output the approximate solution computed For this we
do a simple preorder traversal of the no des in the hierarchy tree HT H and output the explicit no des in
each cell Its easy to see that we can output the solution in O N space since the depth of HT H no
more than n and each no de on a path from ro ot to a leaf is lab eled with a distinct cell and time linear in
the size of E
Time Complexity
3 5
Theorem HVC runs in time O N
Pro of We compute a maximum matching at each level It is well known that a maximum matching for
2 5
a graph G V E can b e found in time O jV j MV Thus computing a maximum matching while
P
k
2 5
are resp ectively the numb er of pins in cells pro cessing G takes O n p time where p p
i i i i i
k 1 l
l=1
which are called in the de nition of G We also compute a maximal matching while pro cessing G G
i i i
k 1
each G and the time for this is O n e where e is the numb er of edges in the level i Therefore
i i i i
P P
n k
2 5
the total time complexity is b ounded by O n O n e which is b ounded by p
i i i i
l
i=1 l=1
3 5
O N
Corollary Given a hierarchical speci cation of a graph G we can compute in time polynomial in the
size of the speci cation the size of an approximate minimum maximal matching which is within a factor
of of the optimal
Pro of Follows from the fact that any maximal matching is within a factor of of the optimal minimum
maximal matching
Approximating Weighted Max Cut
Given an undirected graph G V E the goal of the simple max cut problem is to partition the set V into
two sets V and V such that the numb er of edges in E having one end p oint in V and the other in V is
1 2 1 2
maximized GJ
In HR it is shown that MAX CUT is PSPACE hard In this section we show that given
HG
a hierarchical sp eci cation of a graph G we can compute an approximate max cut which is within
times the optimum and a hierarchical sp eci cation of the vertices in one of the sets in the partition Our
algorithm computes the numb er of edges in the approximate cut in time p olynomial in the size of the
hierarchical description An algorithm for weighted max cut can b e devised along the same lines and is
omitted Since a graph obtained by expanding a hierarchical sp eci cation can in general b e a multigraph
our approximation algorithms treat copies of an edge as distinct edges
We b egin with a brief overview of the algorithm First we recall the idea b ehind the known heuristic
for computing a near optimal weighted max cut in the at non hierarchical case That heuristic referred
to as FMAX CUT in the following discussion pro cesses the no des in arbitrary order and assigns each
no de v either to V or to V dep ending up on which of these sets has edges of least total cost to v As
1 2
in the case of the vertex cover algorithm our approximation algorithm for MAX CUT pro cesses the
HG
b
input sp eci cation in a b ottom up fashion At each level we construct a burnt graph G starting from
i
the original description of the cell G We use the heuristic FMAX CUT to partition the explicit vertices
i
b
at each stage The burnt graph G for G then consists of two sup er no des denoting an implicit partition
i
i
of all the vertices de ned in levels b elow The edges go from a sup er no de to the pins in G Each edge
i
has a weight asso ciated with it The edge weight is the numb er of edges the explicit vertex represented by
the pin has to the vertices in that partition In the following description A denotes the set consisting of
i
all the explicit vertices in G which are not adjacent to any nonterminals in the de nition of G Further
i i
i i
let G A denote the subgraph induced on the no des in A The sets V and V denote the partition of
i i
1 2
i
the vertices of E G Let E denote the numb er of edges in the near optimal cut of E G Also for any
i i
i
vertex v let C ount V denote the numb er of edges having one endp oint as v and the other endp oint in
v
j
i
the set V Throughout this section the reader should b ear in mind that as a consequence of the de nition
j
of hierarchical sp eci cation a terminal an explicit vertex or a pin de ned in G can b e adjacent to at
i
most one pin in each nonterminal called in G The details of the approximation algorithm HMAX CUT
i
app ear in Figure
Example Figure illustrates the execution of the algorithm for the hierarchical sp eci cation given in
Figure The gure consists of columns The rst column corresp onds to G The second column
i
b
denotes the burnt graph G of G As mentioned b efore the weights on the edges denote the numb er of
i
i
i
vertices in V that are adjacent to the pin The third column shows the hierarchical representation H
j
b eing obtained level by level
Pro of of Correctness
We now prove that the algorithm indeed pro duces a valid implicit partition of vertices
Theorem Given a hierarchical speci cation the heuristic HMAX CUT computes a partition of the
given vertex set
Pro of Induction on the numb er of non terminals in the de nition of
Basis When G In this case the theorem follows by the correctness of FMAX CUT
1
Heuristic HMAX CUT
Input A simple hierarchical speci cation G G of a graph G
1 n
n n
Output A hierarchical sp eci cation H H H of the vertices in the set V and E the numb er
1 n
1
of edges in the approximate max cut computed
For i n do
i i
a Use Algorithm FMAX CUT to partition A into sets X and X Note that we do not consider
i
1 2
any edges which are from these explicit vertices to the pins
i i i
b E numb er of edges u v such that u X and v X
1 1 2
in its de nition Let B denote the set of all the explicit G c Let G call nonterminals G
i i i i
m 1
vertices remaining after Step a Note that each of these explicit vertices is adjacent to at
least one nonterminal in the de nition of G We consider the vertices in B one at a time Let
i i
i i
Y Y
1 2
For each vertex v B do
i
v v
i Compute sets V and V de ned by
i i
X X
1 2
i v i v
fw jw X and w is adjacent to v g and V fw jw X and w is adjacent to v g V
i i
1 2
X X
2 1
i v i v
ii If G calls no nonterminals then C ount V jV j and C ount V jV j else
i i
i v v
1 2
X X
1 2
Let v b e adjacent
l m G to pins p
i v i
l l
i i
l l
Then and pin p denote the weight of the edge b etween the sup er vertex V p Let w t V
v i v i
l l
1 1
let
X
i
i v
l
p C ount V jV j w t V
i
v i v
l 1
1
X
1
1lm
X
i
i v
l
p C ount V jV j w t V
i
v i v
l 2
2
X
2
1lm
i i i i i i
iii If C ount V C ount V then Y Y fv g and E E C ount V
v v 2 v
1 2 2 2 2 1
i i i i
else Y Y fv g and E E C ount V
2 v
1 1 2 2
b b
d Construct the burnt graph G as follows The pins in G are the same as the pins in G and we
i
i i
i i
have two sup er vertices V and V which implicitly represent the partition constructed so far Let
1 2
G have m pins in its de nition These pins will b e connected to explicit vertices de ned in G
i i i
is called in the de nition of G Let pin p G b e connected to pin where G and to pins in G
i i i i
r r
i
The weight of an edge p V r m j is calculated as follows in G p
i i
r r
j
X
i
i
r
w t p V jE x G j V w t p
j i i
j r
j
i
r
i i
where E x G X Y denotes the set of explicit no des in G that are connected to p and are
j i i
j j
i
added to V in Steps a and c
j
P
i i i
e E E E E
i
r
1 2
i
r
i
f H has no pins The explicit vertices are in corresp ondence with the vertices in the set X
i
1
i
corresp onding to the nonterminals H Y Furthermore H calls a non terminal of typ e H
i i i
m 1 1
called in G G G
i i i
m 1
S S
i i
j j
i i i i i i
Remark Let V X Y V and V X Y V i i j m where G
j i
j
1 1 1 1 2 2 2 2
i i
j j
app ears in the de nition of G
i
n
Output E and the hierarchical sp eci cation H H H
1 n
Figure Details of MAX CUT Heuristic 0 1 1 0 1 5 1 5 1 VV1 1 2 3 2 3 4
b GGH1 1 1
1 0 0 γ α γ 1 0 2 0 2 VV1 2 HH1 1 G 1G 1 β
b GGH2 2 2
a b G1 c d H1 d b e
1 G G2 H H2 1 e
G H
3 3
Figure Figure showing the execution of heuristic HMAX CUT on the sp eci cation given in Figure
Induction Step Assume that the theorem holds for all sp eci cations with at most n non terminals
By the G G Consider the case when G G G Let G call non terminals G
n n 1 2 n n n
k 2 1
are partitioned induction hyp othesis we know that the the vertices in the hierarchy tree ro oted at G
n
k
n n n n
into two sets The explicit vertices of G are clearly partitioned into two sets X Y and X Y
n
1 1 2 2
S
n
j
n n n
Moreover V X Y r Therefore it follows that the algorithm partitions the V
r
r r r
n
j
vertices into two sets
Performance Guarantee
We rst prove that the weights on the edges in the burnt graph from sup er no des to the pins actually
represent the numb er of no des in the de nition that the pin is adjacent to
Lemma Let be a hierarchical speci cation of a graph G constructed by HMAX CUT Consider the
b
corresponding to the non terminal G in the hierarchical speci cation Then the weight of burnt graph G
i
i
i
an edge from a pin p G to a super vertex V j is equal to the total number of edges from p to
i
j
i
the vertices in the set represented by V
j
i i
Pro of We prove the theorem for V The pro of for V is similar The pro of is by induction on the numb er
1 2
of nonterminals in the de nition of
Basis When G In this case the lemma follows by fact that the weights were calculated by
1
counting the numb er of explicit vertices in G that are adjacent to the pin
1
Induction Step Assume that the lemma holds for all sp eci cations which have no more than n
By the G G non terminals Consider the case when G G G Let G call G
i i 1 2 n n i
m 2 1
induction hyp othesis we know that the lemma holds for the burnt graphs corresp onding to the non
Consider the non terminal G In Steps a and c the explicit vertices G G terminals G
n i i i
m 2 1
n n n n
are partitioned into two sets X Y and X Y Consider a pin p in G Clearly the total numb er of
n
1 1 2 2
P
i
where k m and E dg es G edges from p to the vertices in the set V is equal to jE x G j
p i 1 n
k
j
i
k
n n
E x G X Y represents the explicit vertices in G that are adjacent to the pin p and E dg es G
1 n n i
k
1 1
and are incident on the pin p represents the numb er of edges which have one endp oint in G
i
k
k m pass through the pins Note that the edges incident on the pin p with one end p oint in G
i
k
By the induction hyp othesis the weight represents the numb er of edges from the in the de nition of G
i
k
The lemma now follows pin to the explicit vertices de ned in the graph E G
i
k
We are now ready to prove that the heuristic computes a near optimal maximum cut
Lemma Let be a hierarchical speci cation of a graph G Let denote the number of edges which
j
n
are explicitly de ned in E G Then E
j n
Pro of The pro of is by induction on the numb er of non terminals in the hierarchical sp eci cation
Basis When there is only one non terminal the result follows by the correctness of the pro cedure FMAX
CUT
Induction Step Assume that the theorem holds for all hierarchical sp eci cations which have no more
than n non terminals in their de nition Consider the hierarchical sp eci cation G G G
1 2 n
The edges in E G can b e G G Consider the de nition of the non terminal G Let G call G
n i i n n i
k 2 1
divided into three di erent categories
Typ e edges which have b oth the end p oints explicitly de ned in one of the hierarchy trees ro oted
r k at G
i
r
Typ e edges which have b oth the endp oints explicitly de ned in the de nition of G
n
Typ e edges which have one endp oint de ned explicitly in G and the other endp oint de ned in a
n
r k ro oted at G non terminal o ccurring in one of hierarchy tree HT G
i i
r r
Also let E xp denote the numb er of edges which o ccur explicitly in the de nition of G Then clearly
j j
the total numb er of edges equals
n
X
E xp C r oss
n n i n
r
i
r
where C r oss denotes the set of Typ e edges By induction hyp othesis we know that the vertices in the
n
are partitioned into two sets such that the numb er of edges crossing the cut hierarchy tree ro oted at G
i
k
By Step c explicit vertices in E is at least of the total numb er of edges Therefore i
i r i
r r
G which are not adjacent to any pins are partitioned in such a way that the at least half of the of edges
n
in the subgraph induced by these vertices are cut Each remaining explicit vertex in G is added to the
n
n n
set V or V dep ending on which set has fewer vertices adjacent to it By Lemma the weights on
1 2
the edges from the pins to the sup er no des by give the numb er of no des that the pin is adjacent to in the
n n
hierarchy tree ro oted at that non terminal Therefore E xp C r oss E E and hence
n n
1 2
X
n
E xp C r oss E
n n i n
r
i
r
Theorem Let be a hierarchical speci cation of a graph G Let OPT G denote a maximum cut in
n
E G Then jOPT G j E
Pro of The theorem follows from the ab ove lemma and the fact that jOPT G j
n
Query Problem
n
Using the ab ove hierarchical sp eci cation of the set of vertices in V we can answer the question of which
1
set a given vertex b elongs As mentioned earlier we assume that a vertex v is sp eci ed as a sequence of
nonterminals which o ccur on the path from the ro ot to the nonterminal in which v o ccurs
Theorem Let be a hierarchical speci cation of a graph G with N vertices Given any vertex v in
the graph G we can determine in O N time the set to which v belongs
Pro of Observe that the hierarchy tree HT H of H is identical to HT except that if a no de in HT
is lab eled by G then the corresp onding no de in HT H is lab eled by H This means that the sequence
i i
of nonterminals used to sp ecify v in E can b e directly used to lo cate the nonterminal H in which v
i
may o ccur This implies that given a vertex v one can easily check in O N time if the vertex o ccurs in
H by following the path in the hierarchy tree to the non terminal in which v o ccurs If v app ears in H
n n
then it b elongs to the set V else it is in the set V
1 2
As in the case of vertex cover problem the hierarchical sp eci cation H obtained can b e used to output
n
the V For this we do a simple preorder traversal of the no des in the hierarchy tree HT H and output
1
the explicit no des in each cell This takes O N space and time linear in the size of E
Time Complexity
Theorem The algorithm HMAX CUT runs in time O N M and constructs a hierarchical speci
n
cation of size O N of the set V
1
Pro of Consider the time taken to pro cess G Step a takes O n m time Steps c and d
i i i
take O d O time to pro cess each terminal of degree d in G Therefore the total running time of
j j i
P
O n m Steps c and d is O n m Hence the total running time of the algorithm is
i i i i
1in
O N M Size of each H is no more than n the numb er of vertices in G Hence the size of H is
i i i
P
O n O N
i
i
Approximating Bounded Degree Maximum Indep endent Set
Our heuristic for obtaining a near optimal solution to the maximum indep endent set problem on b ounded
degree hierarchically sp eci ed graphs is based on a well known heuristic in the at case The heuristic in
the at case referred to FIND SET in the subsequent discussion is the following We pick and add an
arbitrary no de v to the approximate indep endent set and delete v and all the no des which are adjacent
to v This step is rep eated until no no des are left It is easy to see that for a graph in which each no de
has degree at most B the indep endent set pro duced by the heuristic is within a factor B of the optimal
value We now show how to extend this heuristic to the hierarchical case Throughout this section we use
V to denote the set of vertices from E G that are in the approximate indep endent set pro duced by the
j j
algorithm The details of the heuristic HIND SET are given in Figure
Performance Guarantee and Pro of Of Correctness
We now show that the approximate indep endent set computed is within a factor of B of the optimal
indep endent set
Lemma The set V produced by HIND SET is a maximal independent set
n
Pro of The pro of follows by an easy induction on the numb er of non terminals in the hierarchical sp eci
cation
Lemma Let OPT G denote the size of an optimal independent set in G E Then jV j
n
OPT (G)
B
Pro of Follows from the fact that every time we cho ose a vertex we delete mark no more than B
terminals explicit vertices and pins
Heuristic HIND SET
Input A simple hierarchical sp eci cation G G of a graph G Each no de of G has a degree of
1 n
at most B where B is a constant
Output A hierarchical speci cation H H H of the approximate indep endent set and jV j the
1 n n
size of the approximate indep endent set
Rep eat the following steps for i n
a Let A denote the set of all the explicit vertices in G Starting from the set A we create a new
i i i
set B as follows For each vertex v A we place it in the set B i v is not adjacent to any of
i i i
the pins marked removed in the burnt graphs of G where G j i app ears in the de nition of
j j
G Let G B denote the subgraph induced on the no des in B
i i i
Remark A vertex v is placed in the set B i none of its neighb ors in G j i have b een placed
i j
in V
j
b Use Algorithm FIND SET on G B to obtain the indep endent set X
i i
Remark We do not consider any edges which are from these explicit vertices to the pins
X
jV j where G j i app ears in the de nition of G c Let jV j jX j
j j i i i
j
Remark V X V where G j i app ears in the de nition of G Observe that the set
i i j j i
j
is created implicitly
b b
d Now construct the burnt graph G for G as follows The pins in G are the same as the pins in
i
i i
G A pin in G is marked removed i the pin is either adjacent to one of vertices in the set X
i i i
or it is adjacent to one of the pins in G j i which is marked removed
j
e Construct H as follows The explicit vertices in H are the vertices in the set X If G calls a
i i i i
non terminal G j i then H calls H
j i j
Output jV j as the size of approximate indep endent set and H H H as the hierarchical
n 1 n
sp eci cation of the approximate indep endent set
Figure Details of Heuristic for Maximum Indep endent Set
Query Problem
As in the case of max cut problem the hierarchy tree of H is identical to the hierarchy tree HT of
except that the corresp onding no des are lab eled by H instead of G
i i
Theorem Let be a hierarchical speci cation of a graph G Given any vertex v in the graph G we
can determine in O N time if v belongs to the approximate independent set obtained
Pro of Given the lab el of any no de as a path in the hierarchy tree it is easy to check if the vertex b elongs
to the indep endent set sp eci ed by H This can b e done by traversing the hierarchy tree HT H and
checking if the vertex app ears in the given H
i
As in the case of previous algorithms we can output the solution in O N space and time linear in the
size of E This can b e done by a preorder traversal of the hierarchy tree HT H
Time Complexity
Lemma The algorithm HIND SET runs in time O N M and constructs an O N size hierarchical
speci cation for the approximate independent set
Pro of The pro of follows by observing that HIND SET pro cesses each of the G in O n m time
i i i
Summarizing the ab ove results we have
Theorem Let be a hierarchical speci cation of a graph G with maximum node degree B Then we
can compute in time O N M the size of the speci cation an approximate independent set which is
within a factor B of the size of a maximum independent set
Approximating Weighted MAX SAT
We now consider the problem of nding a truth assignment to the variables of a hierarchically sp eci ed
instance of SAT so as to maximize the numb er of clauses that can b e simultaneously set to true We
rst outline a heuristic see Figure with p erformance guarantee which works for non hierarchical
sp eci cations of MAX SAT instances The heuristic is a variant of a heuristic for MAX SAT in Jo
We rst observe that the approximation algorithm given in Figure has a p erformance guarantee of
Lemma Let jC j denote the number of clauses in F Let H eu F denote the number of clauses set
true by FMAX SAT Then H eu F jC j
denote the numb er of clauses in the star centered around x We know that the value Pro of Let C
i x
i
P
jC j the lemma follows C clauses Given that assigned to x in Step b satis es at least C
x i x
i i
x
i
Next we show how given a hierarchical sp eci cation of a SAT formula f we can construct a hierarchical
sp eci cation of the bipartite graph corresp onding to f The transformation is given in Figure
It is easy to see that the transformation given in Figure constructs a hierarchical sp eci cation of
the bipartite graph asso ciated with the SAT formula f Thus we have
1 n 1
Lemma Given an instance F F X F X F of SAT Procedure TFORM
1 n 1 n HG
constructs a hierarchical speci cation BG F G G such that
1 n
size of BG F is O siz e F
BG F can be constructed in O siz e F time
E BG F is the bipartite graph associated with the formula E F
The basic idea of the approximation algorithm for the hierarchical case is to mimic the at case
algorithm FMAX SAT The approximation algorithm is fairly simple and its details app ear in Figure
In the rest of the section we let A b e the set consisting of all variables in F which are not adjacent
i i
to any nonterminals in the de nition of F Further let F A denote the subgraph induced on the no des
i i
in A The details of the heuristic HMAX SAT app ear in Figure
i
Pro of of Correctness and Performance Guarantee
The pro of of the fact that the ab ove algorithm guarantees a solution which is within of the optimal value
is easy and follows by verifying the following two lemmas which can easily b e proven by an induction on
the numb er of nonterminals in the de nition of
Lemma Each variable in the SAT formula F speci ed by is assigned a unique truth value
Lemma Let F F F be a hierarchical speci cation of a SAT formula F Consider the
1 2 n
burnt graph corresponding to a non terminal F in the hierarchical speci cation Then the weight of an
i
edge from a pin p to the super vertex P N represents the total number clauses in which the variable
i i i
represented by p occurs un negated negated in the expanded formula denoted by E F
i i
By an easy induction on the numb er of nonterminals in the de nition of and using the ab ove lemmas
we can prove that
Theorem Heuristic HMAX SAT has a performance guarantee of
Query Problem
We show that the algorithm given ab ove can in fact b e used to give a hierarchical description of the truth
assignments to the variables of the SAT formula F
Theorem Let be a hierarchical speci cation of a SAT formula F Given a variable v in the SAT
formula we can tel l in O N time the truth assignment to the variable v
Pro of To do this we simply follow the path from the ro ot to the nonterminal in which the variable o ccurs
and then check the truth value assigned to v Since each clause has at most variables we can also tell
the truth value of any clause in F in O N time
Time Complexity
Theorem Given a hierarchical speci cation of a SAT formula f the algorithm HMAX SAT runs in
P P
O n of the satisfying O n m and constructs a hierarchical speci cation of size time
i i i
1in 1in
assignment to the variables in f such that at least total number of clauses in E are satis ed
Pro of Consider the time to pro cess a cell F If a vertex corresp onding to a variable v has degree d
i j j
in the de nition of F then it takes O d time to nd a truth assignment to v Therefore the total
i j j
running time of Steps b and c is O n m Hence the total running time of the algorithm is
i i
P P
O n m Size of each H is n the numb er of vertices in G Hence the size of H is O n
i i i i i i
1in i
Non Approximability Results
In this section we discuss our results on the non approximability of several natural problems studied in the
literature when instances are sp eci ed hierarchically We show that approximating the numb er of true
gates in a hierarchically sp eci ed monotone acyclic circuit is PSPACE hard We then show that unless P
PSPACE the optimization versions of the high degree subgraph problem and the high vertex and edge
connectivity problems cannot b e approximated to within a factor c
Intuitively problems proven to b e P hard by a local reduction i e a reduction where each gate is
replaced by a corresp onding subgraph or gadget of xed size by a log space reduction from MCVP can
b e shown to PSPACE hard by a p olynomial time reduction from MCVP Such a reduction transforms
HG
the given hierarchical sp eci cation of a monotone acyclic circuit level by level to obtain a hierarchical
sp eci cation of the original problem instance The pro ofs for the non approximability of the optimization
versions of the circuit value problem high degree subgraph problem and the high vertex and edge con
nectivity problems in the non hierarchical case are examples of such local reductions from MCVP This
prop erty of lo cal reduction allows us to lift these reductions to the case when the inputs are sp eci ed
hierarchically
Approximating Numb er of True Gates in MVCP
The Monotone circuit value problem is known to b e PSPACE hard when the circuit is sp eci ed hierarchi
cally LW RH We rst observe that the problem is PSPACE hard even for strongly level restricted
hierarchical sp eci cations
Lemma The problem MCVP is PSPACE hard even for strongly level restricted speci cations in
which a non terminal C cal ls exactly copies of C
i i 1
Pro of Follows from the fact that the instance of MCVP obtained by LW in their reduction from QBF
is of the required form
Before we give the PSPACE hardness pro of for MTG it is instructive to recall the pro of by Serna
HG
Se showing that MTG is P complete The pro of consists of a log space reduction from MCVP Given
0
an instance C of MCVP with n gates the instance C of MTG consists of the same circuit C along with
n
d e additional AND gates forming a chain with the rst element of the chain b eing connected to the
0
output gate of C and the last element of the chain serving as the output for C As the circuit added to
C only propagates the value of output of C it follows that
If C outputs then OPT MTG n
n
e If C outputs then OPT MTG d
It is clear that the reduction can b e done in log space As discussed in Se the result holds even when
instances are restricted to b e planar
We extend this result and show that MTG cannot b e approximated to within any exp onential
HG
function of the optimal To show this the basic idea is to construct a a chain of exp onential numb er
of AND gates using a simple sp eci cation and join this chain in series to the output of an instance of
MCVP
HG
Theorem Unless P PSPACE no polynomial time algorithm can approximate the maximum number
of true gates in MCVP to within any factor of the optimal even for simple strongly level
HG
restricted hierarchical speci cations where denotes the size of the hierarchical speci cation
Pro of Let C fC C C g b e an instance of a simple hierarchical sp eci cation of MCVP in which
1 2 n HG
each C calls exactly two copies of C Let m denote the numb er of gates in C and N denote the size
i i 1
of C We construct an instance D fD D D g of a simple hierarchical sp eci cation ofMVCP
1 2 n HG
2
N
with m gates such that
N
If C outputs then OPT D
2
cN
If C outputs then OPT D for some c
We now discuss the construction of the instance D
Circuit D The Circuit D consists of two disjoint circuits D and D D is identical to C The
1 1 1 1 1 2 1 2 1
circuit D has AND gates connected in series The input of the rst AND gate is connected to two pins
1 1
Similarly the output of the last AND gate is connected to two pins D consists of a series of AND gates
1 3
such that the total numb er of AND gates in D D D equals N n Figure gives a schematic of
1 1 1 2 1 3 1
the ab ove construction
Circuit D i n The Circuit D consists of ve circuits D D D D and D D
i i i 1 i 2 i 3 i 4 i 5 i 4
and D are identical to D The circuits D and D each consists of a single AND gate The AND
i 5 i 1 i 1 i 2
gate corresp onding to D gets its input from two pins and its output is connected to the partial chain
i 1
of AND gates in D The AND gate corresp onding to D gets its input from the partial chain of AND
i 4 i 2
gates in D and its output is connected to a set of pins D consists of a series of AND gates and joins
i 5 i 3
the partial chains of AND gates in the two copies of D The total numb er of AND gates in D D
i 1 i 1 i 2
D equals N n Figure shows the schematic diagram of D
i 3 i i
Construction of D As in D D consists of ve circuits D D D D and D D and
n n 1 n n 1 n 2 n 3 n 4 n 5 n 4
D are identical to D D consists of a series of AND gates and joins the partial chains of AND
n 5 n 1 n 3
gates in the two copies of D The circuits D and D each consists of a single AND gate The input
n 1 n 1 n 2
p ort of the AND gate corresp onding to D is joined to the output p ort of C and the output p ort feeds
n 1
into the partial chain of the AND gates in D The output of the AND gate corresp onding to D is
n 4 n 3
designated as the output of D and the input p orts of D are joined to the partial chain of AND gates
n 3
in D D consists of a series of AND gates such that the total numb er of AND gates in D D
n 5 n 3 n 1 n 2
D equals N n The construction is depicted in Figure
n 3 n
2
Note that the size of D denoted by is O N Now observe that the ab ove construction sp eci es a
circuit in which the output of the circuit corresp onding to C is connected to a exp onentially long chain of
AND gates Given this observation it is not di cult to verify that the following lemma holds
2
cN N
Lemma If the output of C is at least AND gates wil l output a otherwise less than of
those gates wil l output a
Given Lemma and the fact that the ab ove construction of D can b e done in p olynomial time the
theorem follows
Approximating the Ob jective Function of a Linear Program
We now discuss our result concerning the nonapproximability optimizing the ob jective function of a hier
archically sp eci ed linear program The PSPACE hardness pro of consists of it lifting the pro of in Se
showing that approximating the ob jective function of a linear program is log complete for P
Theorem Unless P PSPACE no polynomial time algorithm can approximate the objective function
of an H LP to within any of the optimum even for strongly level restricted simple speci cations Here
denotes the size of the speci cation
Pro of The reduction is from an instance of strongly level restricted simple hierarchical sp eci cation
D fD D D g of the problem MTG We construct an instance of LP F fF F F g
1 2 n HG HG 1 2 n
b ottom up level by level as follows
Construction of F i n Recall that the formula F is of the form
i i
i i i i i
F X X Z f X Z F
i i i
j
j j
1i i
j
X X
d c z
i i i j j
j j
i
i
z 2Z
j
j
where is the ob jective function We now describ e each of the comp onents in the ab ove de nition of
i
F
i
i
The set of dummy variables X is in corresp ondence with the pins of F Note that this implies
i
n
that X
i i i
Z A B where
i
are in corresp ondence with the edges incident on the where the variables in A A A
i i i
r r r
called in D non terminal D
i i
r
i
The set B consists of variables which are in corresp ondence with the explicitly de ned gates
in D and the input p orts of the circuit
i
For the function the co e cients are c and d are all
i i i
i
r
Note This is true b ecause the given circuit sp eci cation is level restricted i X
r
i
i
i
r
Z and the set of variables Z i i called in D we have a call to F Corresp onding to each D
r i i i
r r
i
are in corresp ondence with the set of explicit variables in F which corresp ond to passed to F
i i
r
the explicit gates de ned in D
i
i i
We now describ e the set of inequalities corresp onding to f X Z We have one set of inequalities for
i
each explicit gate in F We also have an additional set of inequalities with each pin that is connected to
i
the output p ort of an explicit gate in D The inequalities are very similar to those given in Se
i
If x corresp onds to an input p ort of the circuit then we have the equation x if the corresp onding
k k
input is and the equation x if the corresp onding input is
k
For an AND gate we have the inequalities x x x x x x x where x is the variable
k j k i k i j k
denoting the AND gate and x x are the variables corresp onding to the gates whose outputs serve
i j
as the inputs for the AND gate If the gate is connected to a nonterminal the variables x and x
i j
corresp ond to the variables that are asso ciated with the edge joining the gate to the nonterminal
For an OR gate we have the inequalities x x x x x x x where x is the variable
i k j k k i j k
denoting the OR gate and x x are the variables corresp onding to the gates whose outputs serve as
i j
the inputs to the OR gate
i
Recall that with each pin we have an asso ciated dummy variable Consider a pin p whose asso ciated
j
i i
dummy variable is x If p is connected to the output p ort of a gate x then we generate the equation
k
j j
i
x x
k
j
For each variable x which denotes an edge going from an explicit gate to a nonterminal i e x
k k
i
is a variable in the set A and is connected to an output p ort of an explicit gate we generate
the equation x x where x denotes the variable corresp onding to the gate which has an edge
k j j
corresp onding to x joined to a nonterminal
k
It is easy to see that the reduction gives rise to a simple strongly level restricted sp eci cation of F
given that D was simple and strongly level restricted Also it is easy to see that the reduction can b e
done in p olynomial time Next observe that the reduction gives rise to a hierarchical sp eci cation F which
represents the set of inequalities which would b e pro duced if the sp eci cation is expanded and Serna s
construction Se applied on the expanded circuit The only di erence that we have some intermediate
2
variables on edges Let N b e the size of D The size of F denoted by is O N
2N
Given the ab ove observations it is easy to verify that the value of is less than if the output of
2
cN
the circuit is and the value of is at least for some c if the output of the circuit is The
theorem follows
Example Consider the hierarchical sp eci cation D as given in Figure The corresp onding sp eci cation
F is given as follows
F x x x x f z x x z x x g
1 1 2 3 4 1 1 2 1 3 4
F f z z z z z g
2 2 3 4 2 3
F a b c d F e f g h
1 1
f z a b z c d g
4 5
f z e f z g h g
5 6
Note that each equation involving an AND or an OR op erator has to b e replaced by the set of inequalities
as discussed earlier
The corresp onding function is also created similarly and is just a sum of all the explicit variables
Observe that the sp eci cation obtained is strongly level restricted and simple
Approximating Connectivity and High Degree Subgraph Problems
Next we consider the problems HVCP HECP and k HDSP when instances are sp eci ed hierarchically
We prove PSPACE hardness results for these problems when instances sp eci ed hierarchically by lifting
the known pro ofs showing the P hardness of the corresp onding problems in the non hierarchical case We
illustrate this idea by presenting the PSPACE hardness pro of for HVCP PSPACE hardness pro ofs for
the other two problems are along the same lines
The pro of given in KSS showing that HVCP is P complete is a log space reduction from MCVP
with additional restriction that outdegrees of all gates and the input no des is at most and there is at
least one input no de with whose value is It can b e easily shown by slightly mo difying the reduction in
LW that
Lemma The problem MCVP is PSPACE hard even for hierarchical speci cations satisfying al l the
HG
fol lowing restrictions
The speci cation is simple
The speci cation is strongly level restricted
Each C cal ls exactly two copies of C
i i 1
The outdegree of al l gates and the input nodes is at most
There is at least one input node with whose value is
The inputs and the outputs al l occur in the last cel l
We recall the construction from KSS to show the P completeness of the HVCP problem Given
an instance C of the MCVP with the restriction that the outdegree of all gates and the input no des is
and there is at least one input no de with whose value is an instance of G HVCP is created as follows
Each input no de of the circuit as well as the output no de is replaced by a K graph as depicted in
2 2
Figure a
Each OR gate of C is replaced by a copy of the graph depicted in Figure e The upp er no des
are called the in nodes and the lower ones are referred to as the out nodes
Each AND gate of C is replaced by a copy of the graph depicted in Figure d
An additional no de v is added and is connected to the out no des of the subgraph used to replace
new
the output gate and all the in no des of the subgraphs replacing the input gates with value The
construction is illustrated through an example in Figure
Using this construction it can b e proven see KSS that the output of C is i the G contains a
connected subgraph As in the previous pro of of PSPACE hardness we lift the reduction in the non
hierarchical case to prove the PSPACE hardness of HVCP
HG
Theorem The problem HVCP is PSPACE hard for simple strongly level restricted hierarchical
HG
speci cations
Pro of We prove the theorem for Given an instance C fC C C g of simple hierarchical
1 2 k
sp eci cation of MCVP in which each C calls exactly two copies of C we construct a simple hier
HG i i 1
archical sp eci cation fG G G g of a graph G such that G has a connected subgraph i the
1 2 n
circuit corresp onding to C outputs a The reduction follows the same outline as in the pro of of Theorem
It is done level by level and at each stage the gates of the circuit are replaced by a gadget dep ending
on whether it is an AND or an OR gate
Graph G Except for a minor mo di cation the graph G is the same as the one obtained using the
1 1
construction given ab ove proving the P completeness of the problem in the at non hierarchical case
The mo di cation is that if a gate in C has its inputs connected to pins then the corresp onding in no des
1
of the graph replacing the gate are also connected to a pair of pins
Graph G i n It has two calls to G corresp onding to the two calls to C in C For each of
i i 1 i 1 i
the explicit gates we replace it by a corresp onding subgraph dep ending on whether it a AND or an OR
gate Again as in G if the input of the gate is connected to pins then the corresp onding in no des are
1
connected to two pins
An example of this construction app ears in Figure The reader should notice that the construction
pro duces a hierarchical description of the graph that would b e obtained if the reduction of KSS were
applied on the circuit pro duced by the expansion E C of the hierarchical sp eci cation C
With the ab ove observations it is easy to see that the following lemmas from KSS hold
Lemma The output of C is i the graph G has a connected subgraph
Lemma The above construction can be done in polynomial time
The theorem now follows from the ab ove lemmas
The pro ofs of the following theorems also follow the same generic pattern as the pro of of Theorem
ab ove The pro of of Theorem lifts the reduction in KSS showing the P hardness of approximating
connectivity and the pro of of Theorem lifts the reduction in AM showing the P hardness of
approximating the high degree subgraph problem
Theorem Unless P PSPACE the optimization version of the problem HVC G and HEC G
HG HG
cannot be approximated to within a factor of c even for simple strongly level restricted hierarchical
speci cations of G
Theorem Unless P PSPACE the optimization version of the problem HDSP cannot be approx
k
imated to within a factor c even for simple strongly level restricted hierarchical speci cations of
G
Conclusions and Related Work
We have presented p olynomial time approximation algorithms with go o d p erformance guarantees for sev
eral natural PSPACE complete problems for hierarchical sp eci cations We have also presented results
concerning non approximability of optimization version of the monotone circuit value problem linear pro
gramming and high degree vertex and edge connectivity problems Our pro ofs of non approximability
can b e extended so as to apply to O log bandwidth b ounded hierarchical sp eci cations where is the
size of the instance obtained after expanding the given sp eci cation The question of whether the high
degree subgraph and high connectivity problems for hierarchical sp eci cations can b e approximated to
some constant factor of the optimal is op en
In MRHR we have shown that e cient approximation algorithms can b e obtained for hierarchically
sp eci ed unit disk graphs In MHR we consider the complexity of nding p olynomial time approx
imation schemes for hierarchically sp eci ed planar graphs In CF a CF a Condon et al give a
characterization of PSPACE in terms of probabilistically checkable debate systems and use this character
ization to show that many natural PSPACE hard problems cannot b e approximated Intriguingly enough
all the problems listed in Table are known to have NC approximation algorithms when the problem
instances are sp eci ed non hierarchically KW PSZ Moreover each of the problems shown to have
a p olynomial time optimal solution in LW a Le Le Wi eg minimum spanning tree planarity
testing when the problem is sp eci ed hierarchically has an NC algorithm when the problem instance is
presented non hierarchically In HM we have shown that for every problem in MAX SNP there is
an NC approximation algorithm A with a constant p erformance guarantee All the problems for which
we have approximation algorithms in the hierarchical case b elong to MAX SNP in the non hierarchical
case While there are problems whose non hierarchical versions can b e solved in NC but their hierarchical
versions are PSPACE hard LW the results here and in LW a Le Le Wi suggest that there is
a strong relationship b etween a problem having an NC algorithm in the non hierarchical case and a p olyno
mial time algorithm in the hierarchical case Understanding this relationship may well lead to a paradigm
for translating known NC algorithms in the literature for problems when sp eci ed non hierarchically to
p olynomial time algorithms for the same problems when the instances are sp eci ed hierarchically
Acknowledgements We thank Venkatesh Radhakrishnan and Richard Stearns for several constructive
suggestions and Lefteris Kirousis for making available the journal version of KSS The rst author also
expresses his thanks to Egon Wanke for fruitful discussions on hierarchical sp eci cations during his visit
to Bonn
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Algorithm FMAX SAT
Input A SAT formula F and its asso ciated bipartite graph
0
Transform the bipartite graph G corresp onding to F into a new bipartite graph G in which the we
have one vertex for each variable one vertex for each clause and if a clause c x y z then we
i
have an edge from vertex corresp onding to c to the vertex corresp onding to x
i
Remark Step intuitively breaks the original bipartite graph into stars with a variable no de as
the center of each star
For each variable x i n do
i
Begin
x x
i i
a Compute the sets PV and NV de ned as
0 x
i
fw j w is a clause no de adjacent to x in G and x app ears unnegated in w g PV
i i
0 x
i
fw j w is a clause no de adjacent to x in G and x app ears negated in w g NV
i i
x x
i i
j then set x to true else set x to false j jNV b If jPV
i i
End
Output The satisfying assignment to the variables of F
Figure A Heuristic for Non hierarchical Instances of MAX SAT C1
D1
C i-1 C i-1
D
i-1 D i-1
Figure Construction of D i n
i 0 1 o/p z2 z 3
z 4 z z 5 6
D D 1 z 1 1
D 1 D 2
1 z 2 0 z 3 o/p
z 4 z 6 z 5
a b c d e f g h
1 2 z 1 z 1
D 1 D 1
E(D)
Figure Example of a circuit represented hierarchically E F represents the actual circuit