Theory of Periodically Specified Problems: Complexity and Approximability

MADHAV V. MARATHE1 HARRY B. HUNT III 2 DANIEL J. ROSENKRANTZ 2RICHARD E. STEARNS2

April 14, 1998

Abstract

We study the complexity and the efficient approx-imability of graph and satisfiability problems whenspecified using various kinds of periodic specifica-tions studied in [Or82a, HT95, Wa93, HW94, Wa93,MH+94]. We obtain two general results. First, wecharacterize the complexities of several basic gener-alized CNF satisfiability problemsSAT(S) [Sc78],when instances are specified using various kinds of1- and 2-dimensional periodic specifications [Or82a,Wa93, HW94, HW95, CM91, CM93]. We outline howthis characterization can be used to prove a num-ber of new hardness results for periodically speci-ifed problems for various complexity classes. Asone corollary, we show that a number of basicNP-hard problems becomeEXPSPACE-hard when in-puts are represented using 1-dimensional infinite pe-riodic wide specifications answering an open ques-tion in [Or82a]. Second, we outline a simple yeta general technique to devise approximation algo-rithms with provable worst case performance guaran-tees for a number of combinatorial problems specifiedperiodically. Our efficient approximation algorithmsand schemes are based on extensions of the ideasin [Ba83,HM85,MH+94] and represent the first non-trivial characterization of a class of problems havingan �-approximation (orPTAS) for periodically spec-ified NEXPTIME-hard problems.

1Current Address: Los Alamos National Laboratory P.O.Box 1663, MS B265 Los Alamos NM 87545. Email:marathe@lanl.gov. The work is supported by the Depart-ment of Energy under Contract W-7405-ENG-36.

2Department of Computer Science, University at Albany- SUNY, Albany, NY 12222. Email addresses of authors:fhunt,djr,resg@cs.albany.edu. Supported by NSF GrantsCCR 90-06396 and CCR94-06611.

1 Introduction

Periodic specifications can be used to define largescale systems with highly regular structures. Usingperiodic specifications, large objects are describedas repetitive connection of a basic module. Fre-quently, the modules are connected in a straightline but the basic modules can also be repeated intwo or higher dimensional patterns. i-dimensionalspecifications were studied by Orlin [Or82a], Wanke[Wa93], Hoppe and Tardos [HT95], Ford and Fulker-son [FF58] and Gale [Ga59]. Two dimensional peri-odic specifications arise naturally in the study of reg-ular systolic arrays and VLSI signal processing arrays[HW94, CS81, IS86, IS86, IS87], discrete dynamicalsystems such as the cellular automata [Wo84], paral-lel programming [HLW92, KMW67], etc. For exam-ple, in the design of Field Programmable Gate Arrays(FPGA’s), the problem of compaction and routing canbe modeled as a shortest path problem in two dimen-sional periodically specified graphs [Br95]. Similarlythe problem of mapping uniform recursive or iterativeprograms on a 2-dimensional mesh connected paral-lel computer is modeled as solving systems of peri-odically specified systems of equations (aka. uniformrecurrence equations) [HLW92, KMW67]. In digitalsignal processing periodic specifications are used todesign bit parallel FIR filter [CS81]. Finally, two di-mensional periodic specifications can also be easilyseen as a way of representing the dynamic changesin the configuration of finite one dimensional cellularautomata over time (i.e the second dimension repre-sents time) [Wo84]. Using this representation the con-figuration reachability problem for a finite one dimen-sional cellular automata is simply the circuit valueproblem for periodically specified circuits. Other re-

1

searchers have studied 2-dimensional and more gen-erally d-dimensional periodic specifications. (See[CM91, IS87, KO91, KS88, Wa93, HW94, HW95].)Typically, the periodic specifications studied in the lit-erature are generalizations of standard specificationsused to describe objects. In general, periodic spec-ifications can describe objects that are exponentiallylarger than size of the specifications themselves.

2 Summary of results

We study the complexity and the efficient approx-imability of graph and satisfiability problems whenspecified using various kinds of succinct specifica-tions with emphasis on various kinds of periodic spec-ifications studied in [Or82a, HT95, Wa93, HW94,Wa93, MH+94]. We present general techniques forproving both hardness results as approximately solv-ing problems so specified. To obtain our results, wesystematically define various kinds of periodic spec-ifications. For uniformity and space reasons, we willconcentrate on generalizedCNF satisfiability prob-lems specified using various types of periodic spec-ifications. The various kinds of periodic specifica-tions considered depend upon the answers to the fol-lowing questions:(1) Is the specified instance1- or2-dimensional ? (2) Is the specified instancefiniteor infinite ? (3) Are specificationsnarrow or wide? (4) Areexplicit boundary conditions allowed inthe specifications ? (5) Are bounds on finite dimen-sions specified inunary (U) or binary (B) ? (6) Dothe infinite dimensions range overnatural numbers(N) or integers (Z)? For the purposes of illustration,we limit our attention in this section to the followingspecifications: (A) The 2-dimensional finite periodicnarrow specifications of Wanke [Wa93], (referred as2-F(B,B)PN-specifications), (B) The 2-dimensionalfinite periodic narrow specifications with explicitboundary conditions (referred as2-F(B,B)PN(BC)-specifications) (C) the 1-dimensional finite wide pe-riodic specifications of Orlin [Or84b] (referred as1-F(B)PW(c) specifications). In Section 3.1 we detailthe naming convention used to specify problems us-ing various kinds of periodic specifications.

2.1 Complexity of Periodically Specified Prob-lems

In the past, succinct specifications were by andlarge studied seperately. Here we present a unifiedframework for obtaining hardness results for prob-lems specified using various kinds of succinct spec-ifications. Roughly speaking, we show that

1. Certain simple repetitive structures can be spec-ified (using small specifications) byall the suc-cinct specifications studied here.

2. We show that basic combinatorial problems are“hard” (for the respective complexity classes)even for such simple repetitive structures spec-ified succintly; thus the problems are “hard”when specified byanyof the succinct specifica-tions studied.

The basic complexity results have a number of appli-cations to proving hardness results including hardnessresults for problems specified using various succinctspecifications, including theS.C.R. andG.C.R. spec-ifications [PY86, BLT92, Ga82]. A complete discus-sion of these applications is deferred to the journalversion of the paper.

We briefly discuss our hardness results for peri-odically specified problems. A summary of our re-sults, for the two problems3SAT and3SATWP spec-ified periodically appears in Table 1. Using the nota-tion of Schaefer [Sc78], all of the hardness results forthe problem3SAT, also hold for each of the prob-lemsSAT(S) andSATc(S) shown to beNP-completein [Sc78]. We can show that efficient reductions in-volving local replacement(possibly augmented withfixed size enforcers) [GJ79] of the problem3SAT 1-3SAT, NAE-3SAT 3SATWP3 etc, to a problem�can be extended to obtain efficient reductions of theproblems3SAT 1-3SAT, NAE-3SAT 3SATWP, etc,to the problem�, when instances are specified us-ing the kinds kinds of periodic specifications consid-ered here. These problems include most of the basic

3Horn formula satisfiability problem is the restriction of theproblem3SAT, in which each clause has at most onepositive lit-eral. This is the same as the problemSATWN, studied by [Sc78].The problem 3SATWP is similar to3SATWN except that eachclause has at most one negated literal.

problems in [Ka72, GJ79] as well as several basicP-complete problems [JL77]. These results yield a num-ber of new hardness results for the complexity classesDSPACE(n), NSPACE(n), DEXPTIME, NEXP-TIME, EXPSPACE etc. depending on the kind ofperiodic specification used. To our knowledge, pre-viously noDEXPTIME, NEXPTIME, EXPSPACE-hardness orundecidability results were known forperiodically specified problems. The periodic lan-guages that can be formalized can be seen as a char-acterization of various space/time complexity classes.For instance 1-dimensional periodic specifications ofOrlin [Or82a] constitute an alternative characteriza-tion of PSPACE. In a similar fashion, the wide 1-dimensional periodic specifications constitute an al-ternative characterization ofEXPSPACE. Our studygeneralizes Orlin’s result [Or82a] that the problem1-I( Z)PN-3SAT is PSPACE-complete and Schaefer’scharacterization [Sc78] of the complexity of gener-alizedCNF satisfiability problemsSAT(S), whereSis a finite set of finite arity Boolean relations. Asone corollary, we prove theEXPSPACE-hardness ofa large class of combinatorial problems when spec-ified by 1-dimensional wide periodic specifications.answering an open question posed by Orlin [Or82a].2.2 Approximation Algorithms

Given the hardness results in the previous sectionfor solving the problems exactly when specified byone of the above specifications, we investigate the ex-istence of polynomial time approximation algorithmsfor these problems. We present a uniform approachfor developing thefirst efficient approximation algo-rithms and/or schemes for a number of optimizationproblems when specified using one of the specifica-tions �. To this end we present a fairly simple yetgeneral technique consisting of two main steps. First,by an extension of ideas in Baker [Ba83] we showthat for each fixed finite setS there is polynomial timeapproximation algorithm (and a scheme for planar in-stances4) for the problemsMAX SAT(S) specified pe-riodically using one of the specifications mentionedearlier in the section. In the next step, we show that anumber of important class of problems when specifiedperiodically can be reduced in an approximation pre-

4corresponding bipartite graphs are planar.

serving way to appropriate problemsMAX SAT(S)specified using the same type of periodic specifica-tions. As a result, we all such reductionsfolded struc-ture preservingL-reductions.

Our main result in this context can be stated as fol-lows:

Theorem 2.1 For each fixed l � 1, and for eachof the problems � listed in Table 1,5 the problem�-�, has a polynomial time approximation algo-rithm with running time O(RT�(l2 � jGj)) with per-formance guarantee6 ( l+1l )2 �FBEST�. Here jGjdenotes the size of the specification, FBEST�denotes the best known performance guaran-tee of an algorithm for the problem � for non-succinctly specified instances and RT�(n) de-notes the running time of the algorithm with in-put size n which guarantees a performance ofFBEST� for the problem �.

As an example, using recent results in [GW94], weget that for all� > 0, the problems2-F(B,B)PN-, 2-F(B,B)PN(BC)-, 2F(B,B)PTN- and 1-F(B)PW(c)-MAX 2SAT have polynomial time approximation al-gorithms that output solutions within a factor of(1 +�) � 1:137 of an optimum solution. As a corollary ofTheorem 2.1, using the recent nonapproximability re-sults of [AL+92] we get the following:

Theorem 2.2 For all the problems � listed in Ta-ble 1, the problems�-� have polynomial time ap-proximation schemes if and only if P = NP.

As a second result which follows from the proofof Theorem 2.1, we get that all of the aboveproblems� have a polynomial time approximationscheme (PTAS), when restricted to planar instances.We can show that many of these problems remainNEXPTIME-complete, even when restricted to pla-nar instances.Theorem 2.3 For all the problems � listed in Ta-ble 1, the problems�-� have polynomial time ap-proximation schemes when restricted to planarinstances.

5In fact we can show that the theorem holds for most problems�-� such that� is in syntacticMAX SNP.6For the sake of uniformity we assume that the performance

guarantee is� 1.

The approximation algorithms have three desirablefeatures:(i) they are conceptually simple, (ii) they ap-ply to large classes of problems�, and (iii) they applyto problems specified using any of the periodic spec-ifications considered here.To our knowledge this isthe first time, polynomial time approximation algo-rithms are developed fornatural NEXPTIME-hardproblems. Thus our results provide the first naturalproblems for which there is a proven exponential (and possibly doubly exponential) gap7 between thetime complexities of finding exact and approximatesolutions8. Only very recently has there even beenwork on the efficient approximability ofPSPACE-hard problems (See [AC94, Co95, CF+93, CF+94,MH+94]). TheNEXPTIME-hardness show that thevery regular structure of problems specified periodi-cally donot suffice to make problems easy. But, theefficient approximation algorithms and schemes de-veloped here show the following:

The very regular structures of problem in-stances specified by the periodic specifica-tions of A-D (mentioned earlier) suffice tomake approximating many basic optimiza-tion problems easy.

Approximating many of the optimization problemsconsidered here, when instances are specified usinghesmall circuit specificationsof [PY86,LB89] can beshown to beNEXPTIME-hard by extensions of thearguments in [Ar94]. Thus our results high-light oneimportant differencebetween multiple-dimension fi-nite periodic specifications and small circuit specifi-cations.

Due to lack of space, the remainder of this paperconsists of preliminary definitions and selected proof

7Previous non-approximability results show that many op-timization problems areNP-hard orPSPACE-hard to approx-imate beyond a certain factor. While these hardness resultspoint out that it isunlikely in general to find “good” polyno-mial time approximation algorithms, they does not rule out thispossibility. The results presented here show aprovable gap be-tween approximation and decision since the decision problemsareNEXPTIME-complete and hence requiresat least2cn stepsandpossibly22cn steps (ifNEXPTIME 6=DEXPTIME) to solve.

8Of course, it is easy to construct artificial problems, whosedecision versions are NEXPTIME-hard, that have polynomialtime approximation algorithms with good performance guaran-tees.

sketches. Additional results are discussed in the Ap-pendix.

3 Preliminary DefinitionsWe first review, the concept of periodically speci-

fied instances. In what follows we discuss the conceptof 2-dimensional periodically specified satisfiabilityproblems. The notion of periodically specified graphsis given in [CM93, CM91, Wa93, HW94, HW95].Figure 1 shows an example of periodic specificationand the associated expanded graph.

For the rest of the paper, letZ andN denote the setof integers and natural numbers respectively.

Let U = fu1; : : : ; ung be a finite set of vari-ables (referred to as static variables).UM;N =fuk(i; j) : 1 � k � n; i 2 f0; 1; 2; � � � ;Mg; j 2f0; 1; 2; : : : ; Ngg. (In our proofs, variableuk(i; j)denotes the variableuk at grid point(i; j).) A literalof U is an element offu1; : : : ; un; u1; : : : ; ung. If wis a literal ofU , thenw(i; j), 0 � i �M and0 � j �N is a literal ofUM;N . LetC(i; j; i+1; j+1) be a pa-rameterized conjunction of 3 literal clauses such thateach clause inC(i; j; i+1; j+1) consists of variablesuk(i; j); uk(i+1; j); uk(i; j+1); uk(i+1; j+1) withthe constraint that at least one variable is of the formuk(i; j). We refer to the clausesC(i; j; i + 1; j + 1)asstatic narrow clauses. (C(i; j) is called narrow be-cause for all(w1(i1; j1) _ w2(i2; j2) _ w3(i3; j3)) 2C(i; j; i + 1; j + 1), jis � irj; jjs � jrj � 1 for1 � r � s � 3.) The conjuction of static nar-row clauses is referred to as static narrow formula.Let � = (U;C(i; j; i + 1; j + 1);M;N). Let C= Vi=M;j=Ni=0;j=0 C(i; j; i+1; j+1). ThenC is the3CNFformula specified by�. GivenUM;N andC, letCM;Nbe a subset ofC with the following property: for eachclause(w1(i1; j1)_w2(i2; j2)_w3(i3; j3)) 2 CM;N ,w1(i1; j1); w2(i2; j2); w3(i3; j3) 2 UM;N .

Definition 3.1 A 2-dimensional finite periodicnarrow specification (2-F(B,B)PN-specification)of a 3CNF formula FM;N (UM;N ; CM;N) is a fourtuple � = (U;C(i; j; i+1; j+1);M;N), where, Uis a a finite set of variables, C(i; j) is a collectionof static narrow 3 literal clauses, and M;N arenon-negative integers specified in binary. Thesize of the specification denoted by size(�) =jU j + jC(i; j; i + 1; j + 1)j + bits(M) + bits(N),

where bits(M) and bits(N) denote the numberof bits used to represent M and N respectively.

The problem 2-F(B,B)PN-3SAT (problem3SAT specified using 2-dimensional finite pe-riodic narrow specifications with both boundsin binary i.e. 2-F(B,B)PN-specifications)is the problem of determining if a 3CNFformula FM;N (UM;N ; CM;N) specified by� = (U;C(i; j; i + 1; j + 1);M;N) is satisfiable.

2-I(Z, B)PN-3SAT is the problem of, givena 2-dimensional periodic specification � =(F (U;C(i; j; i + 1; j + 1);m), where m denotesthe width (in terms of Y-axis) is specified inbinary, determining whether the CNF formula,^i=1;j=mi=�1;j=0C(i; j; i + 1; j + 1) satisfiable.

3.1 Note on Naming Convention

Since we have a large number of parameters, it isnecessary to state the notation used throughout thisabstract for naming problems. We useF andI to de-notefiniteor infinitegraphs respectively. Observe thatwhile this is the property of the expanded object, wechoose to use this as a way to classify the specifica-tion itself. The symbolsU, B in the brackets follow-ing F specify, whether the finite bounds are specifiedin unary or binary notation. The symbolsN, Z fol-lowing I specify whether the graph is infinite in onedirection or both the directions. We have already ex-plained the concept of narrow and wide specifications.We useN andW to denotenarrow andwide spec-ifications respectively.Dimensions of the expandedGraph: f1; 2; : : : dg denote the dimensions in whichthe static graph is translated. Some instances of prob-lems arising in practice have a periodic specificationof the graph or a formula along with explicit initialand final conditions. We call such periodic specifica-tions as periodic specifications with boundary condi-tions (BC).Example 1: Let the set of static variablesU =fx; y; zg. The static clausesC is specified byC(i; j; i + 1; j + 1) = [x(i; j) + y(i; j) + z(i; j)] ^[x(i+1; j)+y(i; j)+z(i+1; j)]^[x(i; j+1)+z(i; j)].The set of clausesC1;1 is given by[x(0; 0) + y(0; 0) + z(0; 0)] ^ [x(0; 1) + y(0; 1) +z(0; 1)] ^ [x(1; 0) + y(1; 0) + z(1; 0)] ^ [x(1; 1) +y(1; 1) + z(1; 1)]V[x(1; 0) + y(0; 0) + z(1; 0)] ^

[x(1; 1) + y(0; 1) + z(1; 1)] ^ [x(0; 1) + z(0; 0)] ^[x(1; 1) + z(1; 0)]4 NEXPTIME-completeness of2-F(B,B)PN-

3SATThe main idea involves the construction of a static

formulas, force the satisfiability of the expanded for-mulas to correspond to the existence of legal com-putations of Turing machines. Intuitively, we haveone column for each step of a computation and onerow for each tape cell of the Turing machine. Prov-ing hardness results for satisfiability problems with-out explicit boundary conditions is subtle, since thereis no obvious way to force the the Turing machine tostart correctly.

Theorem 4.1 2-F(B,B)PN-3SAT is NEXPTIME-complete.

Proof Sketch: Membership inNEXPTIME followseasily by observing that the size of the expanded for-mula is 2c(jU+C(i;j;i+1;j+1)j+bits(M)+bits(N)), where� = (U;C(i; j);M;N), is the specification ofFM;N . Hence aNEXPTIME bounded TM canguess an assignment to the variables and then ver-ify in DEXPTIME that the assignment satisfies all theclauses.

Next, we discuss the reduction which shows theNEXPTIME-hardness of the problem. It is worthpointing out the basic technique used behind the re-ductions. Since the static formula associated with2-FPN-3SAT instance is the same for each time pe-riod, it is not possible to write a3CNF formula whichsays that the machine has the correct startingID. Thismakes the task of constructing the3SAT instancemore difficult. In order to overcome this difficulty, ourreduction consists of two phases. In the first phase,we start with a given Turing machine� with inputx = (x1; : : : ; xn) and construct a new Turing ma-chine�x which simulates� on x and has the follow-ing additional properties that

1. If Turing machine� does not acceptx, then ev-ery possible computation of�x halts within2c0nmoves, else

2. If Turing machine� acceptsx, then�x has acycling computation, where the length of anIDnever exceeds2d0n, for some givend0.

The second phase consists of constructing an in-stance(Ux(t; y); Gx(t; y; t+ 1; y + 1);M;N)) of 2-FPN-3SAT by a polynomial time reduction from�x.Now we know that eachID of the Turing machine�xis of length2d0n + 1. From Property 2 above, weneed to consider only2d0n different ID’s for our re-duction. In order to understand the construction imag-ine eachID of the Turing machine�x being placedvertically in the plane. Two consecutiveID’s of �xare placed vertically next to each other. For the sakeof exposition we will refer to the X-axis as thetimeline. In the following discussion, each grid point isreferred to as(t; y). We now define the set of vari-ablesUx(t; y) and their intended meaning.Ux(t; y)consists of the following three different types of vari-ables. (i)TAPE � Ux(t; y), such thatTAPE(t; y)encodes theyth symbol in thetth ID. TAPE(t; y)takes values from the setf#g [ � [ (Q� �), where� denotes the tape symbols andQ denotes the set ofstates of�x. The number of variables needed to en-codeTAPE(t; y) depends only on the machine�x.(ii) In order to simulate the behavior of�x properlywe need to have two set of counter variables;cy andct. The countercy keeps track of the particular tapecell in a givenID. Let q = d0n. The counter canbe simulated by means of (d0n + 1) Boolean vari-ablestcq; tcq�1; : : : tc0. tc0 represents the least sig-nificant bit andtcq represents the most significantbit. The counterct keeps track of the number ofID’s. The counterct can be simulated by means of(d0n + 1) Boolean variables. We use Boolean vari-ablesyc0; yc1; yc2; : : : ; ycq to simulate the countercy. (iii) Auxiliary variables for making the resultingstatic formula narrow and in the3CNF form.

The initial ID is of the form#(q0; x1) : : : xnB2d0n�n, where B denotes ablank. The static formulaCNF formulaGx(t; y) isgiven byGx(t; y) = f1(t; y)^ f2(t; y)^ f3(t; y). Wenow describe each of the subformulasfi; 1 � i � 3separately. Eachfi is described in terms of variablesat coordinatesy; y + 1; t; t + 1. Counter Updatingformula f1 is used to simulate a counter to achieveimplicit initialization. Implicit Initialization for-mula f2 can be thought of as a way to implicitlyinitialize the clauses to reflect that the machine startsout right whenever the counters are reset to 0. The

initialization condition say that if both the countervalues are 0, then we have # as the tape symboland similar such statements decribing the initialconfiguration. Consistency Checking formulaf3ensures the consistency of the tape symbols, i.e. thatthe contents of the tape cellsi, i + 1 and i + 2 inIDt determine the contents of the tape cellsi, i + 1andi + 2 in IDt+1. The details of these formulas areomitted from the abstract.

We now prove the correctness of our reduction. Ifthe Turing machine� acceptsx then we know that�x has a cycling computation. Hence by setting thecountersct(0; 0) = cy(0; 0) = 0 we get that thefirst column of the grid contains the right initial ID.From then on, the consistency conditions ensures thatthe formula

Vy=N;t=My=0;t=0 Gx(t; y; t + 1; y + 1) is sat-isfied. Conversely, assume that the formula is satisfi-able. SinceM andN are suitably large integers, it isguaranteed that the following two conditions hold:

1. SinceN is large enough, the simulation mustbe carried out for enough steps so that the Tur-ing machine�x goes through the sequencect =0; ct = 1; ct = 2; � � � ct = 2d0n. This impliesthat the formulasf2(t; y) andf3(t; y) would betrue from the time when the value ofct = 0.

2. Similarly, sinceM is large enough, the grid issufficiently long in theY-direction so that thecounter valuecy goes through a sequence of val-uescy = 0; cy = 1; cy = 2; � � � cy = 2d0n. Thisimplies that the first part of the implication inf2is true and from then on, it is ensured that theTM �x goes through the simulation correctly.

The above two conditions imply that if the formulaVy=N;t=My=0;t=0 Gx(t; y; t + 1; y + 1) is satisfied then theTuring machine� acceptsx.4.1 Proving EXPSPACE-hardness of 2-

I(N,B)PN-3SATAlthough there are technical difficulties, the basic

idea behind the proof is similar to the idea used toprove NEXPTIME-hardness of2-F(B,B)PN-3SAT.Therefore, we only point out essential differences.Recall that we used two counters to keep a track of thelength of eachID and also to keep track of the numberof ID’s. Since in the proofs ofNEXPTIME-hardness,we need only consider singly exponential manyID’s

we were able to use a counter which had only polyno-mial number of bits. This in particular implied that thevariables constituting the counter can occur togetherexplicitly in the static formula. In this case, we wantto simulate an2cn space-bounded Turing machine andthis means that we need to keep track of roughly22cnID’s. To do this we need a counter with roughly2cnbits. Thus all the variables constituting the countercannot occur togetherexplicitly in the static formula.To get the intuition, It is useful to imagine eachID be-ing rotated horizontally on theX-axis. Now observethat the narrow clauses used to describe the relation-ship between variables of consecutiveID’s in case of2-dimensional specifications can be replaced by wideclauses in the 1-dimensional specifications.

4.2 Polynomial time solvability of 2-F(Z,Z)PN-3SATWP

Next, we consider the problems2-F(B,B)PN-3SATWP, 2-I(N,N)PN-3SATWP and 2-I(Z,Z)PN-3SATWP. Extending our results for these prob-lems to similar problems involving Horn formulasatisfiability is straightforward and is omitted here.In contrast to theundecidability of solving 2-I(N,N)PN(BC)-3SATWP, we show that each of theabove three problems has apolynomial time algo-rithm. This points out a major difference betweenthese variants of periodic specifications.

We first consider the problem2-I(Z,Z)PN-3SATWP. Recall that a relationR is weakly posi-tive if R is equivalent to someCNF formula having atmost one negated variable in each conjunct. The algo-rithm for solving the problem2-I(Z,Z)PN-3SATWPis relatively easy, and is based on the following twoobservations. The first observation is that if there isa clause with only one literal, all copies of the cor-responding variable must have the same value. Forinstance, if there is a clause consisting of the singleliteral xi(t+ 1; y), then all copies of variablexi haveto be set to false. The second observation is that af-ter simplifying the set of clauses as much as possi-ble on the basis of the first observation, every remain-ing clause has either no literals or more than one lit-eral. Weak positivity implies that each clause withmore than one literal contains at least one positiveliteral, so setting all remaining variables to true willsatisfy all such clauses. Since each simplification of

the set of clauses based on the first observation as-signs a value to a variable in the static formula whichhas not been previously assigned a value, the algo-rithm will terminate in polynomial time. Note thatif the expanded formula for the given instance of2-I(Z,Z)PN-3SATWP is satisfiable, there exists a sat-isfying assignment that assigns the same value to allthe copies of a given variable in the static formula.

Next consider the problems 2-I(N,N)PN-3SATWP and 2-F(B,B)PN-3SATWP. Any al-gorithm for solving these problems must deal withsubtle issues created by the presence of a “boundary”in the expanded formula. A clause of the formxi(t; y) implies thatxi is set to true for all timeperiods. However, a clause of the formxi(t + 1; y)does not imply anything about the value of thevariable xi(0; y) in a satisfying assignment of theformula. Similar arguments hold for clauses of theform xi(t; y + 1) andxi(t + 1; y + 1) (the secondclause might arise after the elimination of othervariables.) The following simple example, shows thateven for 1-dimensional specifications, there are caseswhere all satisfying assignments to the expandedformula assign different values to the copies of aparticular variable.

Example 2: LetF = (U;C(t; t+1); 1) be an instanceof 1-F(B)PN-3SATWP where the set of static clausesare given by(x1(t) + x2(t+1))^ (x2(t))^ (x2(t) +x1(t+ 1)). The set of variables areU = fx1; x2g.The expanded formula is(x1(0)+x2(1))^(x2(0))^(x2(0) + x1(1)) ^ (x2(1)):

By inspection it is clear that any assignment tothe variables of the expanded formula such thatv[x1(0)] = v[x1(1)] andv[x2(0)] = v[x2(1)] cannotsatisfy the formula. But the assignmentv[x1(0)] = 1,v[x1(1)] = 0, v[x2(0)] = v[x2(1)] = 0 satisfies theexpanded formula. A similar example can be con-structed for the 2-dimensional case.

Example 2 suggests that a polynomial time algo-rithm for solving 1-I(N,N)PN-3SATWP should dis-tinguish between the copy of each variable att =0; y = 0 and the copies of the same variable at timet; y � 0. Our polynomial time algorithm for2-I(N,N)PN-3SATWP considers four groups of vari-ables, corresponding to(t = 0; y = 0), (t = 0; y >0), (t > 0; y = 0), and(t > 0; y > 0).

5 Approximation Algorithms for 2-F(B,B)PN-specified problems

The basic idea behind our approximation algo-rithms involves the conversion of solutions obtainedfrom alocal algorithm on small sub-grids to a solutionof theglobal problem. The method of partial expan-sion involves the application of a divide and conqueralgorithm iteratively by considering different subsetsof the given graph; solving each subset by a localalgorithm, constructing a global solution and finallychoosing the best solution among these iterations asthe solution to�. The method can be seen as an exten-sion of the shifting strategy devised by Baker [Ba83]for finding efficient approximation algorithms for sev-eral combinatorial problems.

We outline the basic technique by discussing ourNC-approximation scheme for the maximum inde-pendent set problem. Consider a2-F(B,B)PN spec-ification of a graphG, and an integerk > 1. Tobegin with, for eachi, 0 � i � k, we partitionthe graphG into l disjoint setsG1; � � �Gl by re-moving vertices with horizontal coordinates congru-ent to i mod (k + 1). For each subgraphGp,1 � p � l, we find an independent set of size atleast kk+1 times the optimal value of the independentset inGp. The independent set for this partition isjust the union of independent sets for each ofGp.By an averaging argument, it follows that the parti-tion which yields the largest solution value containsat least( kk+1)2 �OPT (G) nodes, whereOPT (G) de-notes the value of the maximum independent set inG. (For simplicity, we use a symbol to denote a set aswell as its cardinality. The intended meaning will beclear from the context.)

It is important to note that the size of the graph weare dealing with is in general exponential in the sizeof the specification. Hence a naive application of theabove idea will lead to algorithms that take an expo-nential amount of time. However, as we shall see, the“regular” structure of the graph allows us to solve theproblems considered here in time polynomial in thesize of the specification.

We illsutrate the basic technique by discussing analgorithm for solving2-F(B,B)PN-MAX-IS problem( ALGORITHM ALGORITHM ALG-2-FPN-MAX-IS). The correctness and the performance guarantee

through a series of intermediate results discussed be-low.5.1 Approximation Scheme for 2-F(B,B)PN-

MAX-IS: PerformanceLemma 5.1 For each iteration of loop 2(c)i, thegraphs Gi1;jki;j , 2 � j � r � 1; 2 � jk � sj �1 (i.e. the graphs Gi1;2i;2 , Gi1;3;i;r�1 � � � Gi1;sj�1i;r�1 ) areisomorphic.

Proof Idea: Follows from the definition of periodicspecification.

Let us define two subgraphs obtained in iteration2.(a).i.A to be in the same equivalence class if theyare isomorphic. Then it is easy to see that the maxi-mum independent set problem need only be solved forexactly one member of each equivalence class. As acorollary of the above lemma and by definition of pe-riodic specifications we get that the number of equiv-alence classes are finite. Furthermore, as result of ourpartitioning step, it can be shown that the size of theindividual pieces isO(k2 � jGj). These crucial factsallow us to bound the running time of our algorithmbyO(RT�(k2 � jGj)).Lemma 5.2 The number of equivalence classesis no more than 9. Furthermore, The numberof elements in each equivalence class is a poly-nomial time computable function f (in the sizeof the specification) of M and N , denoted byf(M;N).Proof Idea: Foolows by definition of periodic speci-fication.

Lemma 5.3 Each of the subgraphs Gi1;j1i;j ob-tained in Step 2.(c).i.B is disjoint.

Proof Sketch: Follows from the property of instancesspecified by2-F(B,B)PN specifications; namely avertex defined at grid point(i; j) is adjacent only tovertices that are defined at grid points(l;m) such thatjl � ij; jm� jj � 1.

Next, we prove that the algorithm given above in-deed computes a near optimal independent set. Thatis, given anyk > 1 the algorithm will compute an in-dependent set whose size is at least( kk+1)2 times thatof an optimal independent set.

First, we prove that of all the different iterationsfor i, at least one iteration has the property that thenumber of nodes that arenot considered in the in-dependent set computation is asmall fraction of anoptimal independent set.

Recall that for eachi we did not consider the ver-tices which were placed at lattice points with horizon-tal coordinatesj1; j2 � � � jp such thatjl � i mod(k +1), 1 � l � p. LetS0; S1; � � � Sl be the set of verticeswhich werenot considered for each iterationi. LetISopt(Si) denote the vertices in the setSi which werechosen in the optimal independent setOPT (G).Lemma 5.4max0�i�k jOPT (Gi)j � ( k(k + 1)jOPT (G)jProof: The proof follows by observing that the fol-lowing equations hold:0 � i; j � l; i 6= j; Si\Sj = �; [t=lt=oSt = V (G):

The proofs of the theorem follows by an averagingargument. We omit the proofs due to the lack of space.

Theorem 5.5 jIS(G)j � ( kk+1)2 � FBEST �jOPT (G)j. Here FBEST denotes the perfor-mance guarantee of the best algorithm known tosolve the independent set problem.

AcknowledgementsWe thank A. Condon, J. Feigenbaum, T. Lengauer,

J. Orlin, S. Shukla, R. Sundaram, S.S. Ravi and E.Wanke for valuable discussions on succinct specifi-cations, and the anonymous referees for constuctivecomments and suggestions. Thanks to J. Balcazar,van Emde Boas, M. Furer, J. Orlin and E. Cohen formaking available copies of their papers.

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u(0,0)

v(0,0)

w(0,0)

u(1,1)

v(1,1)

w(1,1)

u(0,1)

v(0,1)

w(0,1)

u(1,0)

v(1,0)

w(1,0)

u(2,0)

v(2,0)

w(2,0)

u(2,1)

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v(2,1)(1,0)

(1,0)

(1,0)

(1,0)

(1,0)

u

v

w

(0,1)

(a) (b)

Figure 1: (a)The static graph with 2-dimensional integer vectors associated with each edge. (b) The graphG2;1specified by� = (G; 10; 01).

ALGORITHM ALGORITHM ALG-2-FPN-MAX-IS� Input: An instance (G;M;N ) of a periodic graphGM;N and an� > 0� Output: A periodic specification of a near optimal independent set inGM;N whose size is at least(1��)2 �FBEST �OPT whereOPT is the size the maximum independent set inGM;N andFBESTdenotes the best possible factor achievable by any polynomial time approximation algorithm for themaximum independent set problem specified using a standard specification.� 1. Letk = d1=�e � 1.

2. For eachi, 0 � i � k do

(a) Partition the graph intori disjoint setsGi;1 � � �Gi;ri by removing all the vertices at gridpoints with X-coordinate congruent toi mod (k + 1).

(b) Gi = S1�j�r Gi;j(c) For eachj, 1 � j � ri do

i. For eachi1, 0 � i1 � k doA. Partition the graphGi;j into sj disjoint setsGi1;1i;j � � �Gi1;sji;j by removing vertices at

grid points with Y-coordinate congruent toi1 mod (k + 1).B. Gi1i;j = S1�j1�sj Gi1;j1i;jC. For eachGi1;j1i;j , 1 � j1 � sj compute the optimal (near optimal) value of the inde-

pendent set denoted byIS(Gi1;j1i;j ).Remark: This can be done by running the algorithm on just three graphsnamely;Gi1;1i;j ,Gi1 ;2i;j andGi1;sji;j

D. IS(Gi1i;j) = S1�j1�sj IS(Gi1;j1i;j )(d) IS(Gi;j) = max0�i1�k IS(Gi1i;j)(e) IS(Gi) = S1�j�ri IS(Gi;j)

3. IS(G) = max0�i�k IS(Gi)

Problems Specification TypeType Name Dimension(s) Finite (F) Boundary Edges F/I Complete

X,Y or Conditions Narrow (N) BoundsInfinite (I) Present or

Yes (Y) Wide (W) F,Unary (U)or No (N) F, Binary (B)

I, Int. (Z)I, Nat. (N)

1-F(B)PN(BC)-3SAT 1 F Y N B NSPACE(n)-C1-F(B)PN-3SAT 1 F N N B NSPACE(n)-C1-I(Z)PN-3SAT 1 I N N Z NSPACE(n)-C1-I(N)PN-3SAT 1 I N N N NSPACE(n)-C

1-F(B)PW(BC)-3SAT 1 F Y W B NEXPTIME-C1-F(B)PW(BC)-3SAT 1 F N W B NEXPTIME-C

1-I(ZPW-3SAT 1 I - W Z EXPSPACE-C2-F(B, U)PN-3SAT 2 F,F - N,N B, U NSPACE(n)-C

3SAT 2-F(B, B)PN-3SAT 2 F,F - N,N B, B NEXPTIME-C2-I( Z, B)PN-3SAT 2 I,F - N,N B, Z NEXPTIME-C2-I(Z, Z)PN-3SAT 2 I,I - N,N Z,Z undecidable2-I(N, N)PN-3SAT 2 I,I - N,N N,N undecidable

1-F(B)PN-3SATWP 1 F N N B Poly1-I(Z)PN-3SATWP 2 I N N Z Poly

2-I(Z, Z)PN-3SATWP 2 I N N Z,Z Poly2-I(N, N)PN-3SATWP 2 I N N N,N Poly

3SATWP 1-F(B)PN(BC)-3SATWP 1 F Y N B DSPACE(n)-C1-F(B)PW(BC)-3SATWP 1 F Y W B DEXPTIME-C

2-F(B, U)PN(BC)-3SATWP 2 F,F Y N,N B,U DSPACE(n)-C2-F(B,B)PN(BC)-3SATWP 2 F,F Y N,N B,B DEXPTIME-C2-F(Z, B)PN(BC)-3SATWP 2 I,F Y N,N Z,B EXPSPACE-C2-F(N, N)PN(BC)-3SATWP 2 I,I Y N,N N, N undecidable

Table 1:Table summarizing the results for the problems3SAT and3SATWP when instances are specified using variouskinds of periodic specifications. For example, the 8th row in the tablespecifies that the problem3SAT when specified using2-dimensional finite periodic narrow specifications, with the boundson theX-axis specified in binary and the bounds on theY-axis specified in unary isPSPACE-complete.Z, N stand for integers and natural numbers respectively.

2-F(B,B)PN StandardSpecs Specs

Problem Planar Arbitrary Planar Arbitrary

MAX-3SAT ( l+1l )3 ( l+1l )2 � 4=3 ( l+1(l ) 4=3MAX-SAT(S) ( l+1l )3 ( l+1l )2 � 2p ( l+1(l ) 2p

MIN-VC ( l+1l )3 ( l+1l )2 � 2 ( l+1l ) 2MAX-IS ( l+1l )3 ( l+1l )2 � b ( l+1l ) b

MIN-DOM-SET ( l+1l )3 ( l+1l )2 � b ( l+1l ) log bMAX ED- DOM-SET ( ll�1 )3 ( ll�1 ) 2

MAX-PART- INTO-TRIAN ( l+1l )3 ( l+1l ) � 3 ( l+1l ) 3

MAX-H- MATCH ( l+1l )3 ( l+1l ) � (jVH j=2 + �) ( l+1l ) (jVH j=2 + �)MAX-CUT ( l+1l )2 ( l+1l ) � 1:137 polynomial 1:137

Table 2: Performance Guarantee Results for Optimization Problems for problems specified using 2-F(B,B)PN-specifications. Problem names are clear from the abbreviations. All the problems can be shown to beNEXPTIME-hardusing the method outlined in the paper. Similar results hold for problems specified using2-F(B,B)PN(BC) 2-F(B,B)PTNand1-F(B)PW(c)-specifications respectively.b denotes the degree bound.p denotes the maximum arity of a relation inS.The approximation results for the standard case for arbitrary and planar instances can be found in [CK94].