The Complexity of Approximating PSPACEComplete Problems ;; for Hierarchical Sp ecications Madhav V Marathe Harry B Hunt III SS 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 ecied graphs many of which are PSPACEcomplete Assuming P 6 PSPACE the exis tence or nonexistence of such ecient approximation algorithms is characterized for several standard graph theoretic and combinatorial problems We present p olynomial time approximation algorithms for several standard PSPACEhard 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 ecied hierarchicall y We present p olynomial time approximation algorithms for the following problems when the graphs are sp ecied 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 ecied hierarchicall y the number of true gates in a monotone acyclic circuit when al l input values are specied 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 ecied hierarchicall y high degree subgraph k vertex connected subgraph and k edge connected subgraph Classication Hierarchical Sp ecications Approximation Algorithms Computational Complexity Al gorithms and Data structures 1 Email addresses of authorsfmadhavhuntravigcsalbanyedu 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 ecication and analysis Other applications of hierarchical sp ecication and consequently of hierarchically sp ecied graphs are in the areas of nite element analysis LWb 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 dened by Lengauer in HLW Le Le LW to describ e graphs Using this mo del Lengauer et al Le LWa Le have given ecient 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 ecied graphs including PSPACEcomplete 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 ecied problems is motivated by the following two facts (n) n size hierarchical sp ecications can sp ecify size graphs Many basic graph theoretic prop erties are PSPACEcomplete 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 ecied hierarchically What we mean by a polynomial time approximation algorithm for a graph problem when the graph is sp ecied 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 ecication of a graph G We provide ecient algorithms for the following versions of the problem The Approximation Problem Compute the size of a nearminimum 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 denition 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 ecication 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 ecication rather than the size of the graph obtained by expanding the sp ecication 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 ecication This is a natural extension of the denition of approximation algorithms in the at ie nonhierarchical 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 died 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 ecication and thus O pol y log in the size of the sp ecied graph when the size of the graph is exp onential in the size of the sp ecication 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 ecication 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 ecied graphs We b elieve that this is the rst time ecient approximation algorithms with go o d p erformance guaran 4 tees have b een provided b oth for hierarchically sp ecied problems and for PSPACEcomplete 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 CFa CFa have investigated the approximability of other PSPACEcomplete 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 PSPACEhard Loghard for of True Gates to approximate P to approximate in a circuit for any for any Optimal Value of PSPACEhard Loghard for Ob jective Function for P to approximate of a Linear Program any for any High Degree PSPACEhard Loghard for Subgraph for P to approximate for k Vertex PSPACEhard Loghard for Connectivity for P to approximate for k Edge PSPACEhard Loghard 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 Denitions and Description of the Mo del The following two denitions are from Lengauer Le Denition A hierarchical specication 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 dened 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 i1 of the typ e G then the terminal vertices which are the neighb ors of G are in onetoone 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 Denition Let G G be a hierarchical specication of a graph G The expansion E 1 n ie the graph associated with of the hierarchical