Probabilistically Checkable Debate Systems and Approximation Algorithms for PSPACE-Hard Functions (Extended Abstract)* Anne Condont Joan Feigenbaum$ Carsten Lund~ Peter Shorl~ Abstract 1 Introduction Suppose that two candidates, B and C, agree to a de- bate format. Voter V is too busy to catch more than We initiate an investigation of probabilistically check- a very small number of bits of the debate. How does able debate systems (PCDS’S), a natural generalization V decide which of B or C won the debate? In this pa- of the probabilistically checkable proof systems stud- per, we show that if B and C choose the right debate ied in [1, 2, 3, 8]. A PCDS for a language L consists format, V’s problem is solved. By listening to a few, of a probabilistic polynomial-time verifier V and a de- randomly chosen, sounds bites of the debate, V can bate between player 1, who claims that the input z is with near certainty figure out who won. in L, and player O, who claims that the input x is not Similarly, suppose that B or C is giving a speech in L. We show that there is a PCDS for L in which to a set of voters VI, . Vn, represented by finite au- V flips O(log n) random coins and reads O(1) bits of tomata. He would like to give the speech that results the debate if and only if L is in PSPACE. This char- in acceptance (votes) by the greatest number of Vi ‘s, acterization of PSPACE is used to show that certain PSPACE-hard functions are as hard to approximate We show that not only can he not compute this maxi- mum exactly, but he cannot come within an arbitrary as they are to compute exactly. constant factor, unless he has access to an oracle (po- litical consultant) with the full power of P!3PACE. Our work builds on the recent progress that has been made in the theory of probabilistically checkable t University of Wisconsin, Computer Sciences Depart- proof systems (PCPS ‘s). Results about the language.- ment, 1210 West Dayton Street, Madison, WI 57306 USA, recognition power of PCPS’S have led to lower bounds condon@cs. wise. edu. Supported in part by NSF grants CCR- on the difficulty of approximating NP-hardl functions. 9100886 and CCR-9257241. In this paper, we define probabilistically checkable de.. tAT&T Ben Laboratories, Room 2C473, 600 Mowt~n Avenue, P. O. Box 636, Murray Hill, NJ 07974-0636 USA, bate systems (PCDS’S). We prove several results about jf~research. att . corn. the language-recognition power of PCDS ‘~s and then ~AT&T Ben Laboratories, Room 2C324, 600 Momt& use them to obtain lower bounds on the difficulty of Avenue, P. O. Box 636, Murray Hill, NJ 07974-0636 USA, approximating PSPACE-hard functions. lrmd@research. att. coin. ~AT&T Bell Laboratory=, Room 2DI 49, 600 Momtain Let us describe the background for this work in more Avenue, P. O. Box 636, Murray Hill, NJ 07974-0636 USA, detail. Loosely speaking, a language L has a PCPS if, shor~research. att. corn. for every z c L, there is a string m such thi~t a proba- * The full version of this paper has been submitted for journal bilistic verifier V can be convinced with high probabil- publication and is available as DIMACS TR 93-1o. ity that z c L. The class PCP(r(n), g(n)) consists of Permission to copy without fee all or part of this material is those languages recognizable by PCPS’S in which the granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the verifier uses O(r(n)) coin flips and looks at O(q(n)) title of the publication and its date appear, and notice is given bits. It was recently shown that PCP(log.-, n, 1) = NP that copying is by permission of the Association for Computing (cf. [1, 2]). Machinery. To copy otherwise, or to republish, requires a fee Results on the power of classes PCP(r(n], g(n)) can and/or specific permission. 25th ACM STOC ‘93-5 /93/CA, USA be used to show that many important approximation 01993 ACM 0-S9791-591 -7/93 /0005 /0305 . ..$1.50 305 problems are hard, unless there is some unexpected The following is a technical building block, interest- collapse of complexity classes. The first result along ing in its own right, that is used in the proof that these lines showed that MAX-CLIQUE was difficult to PSPACE = PCD(log n, 1): If r(n) = fl(log n), then approximate [8]. The seminal result of [8] haa been im- PCD(r’(n), q(n)) contains the same languages if the proved several times, and it is now known that there is verifier reads O(q(n)) rounds of the debate as it does an c such that approximating MAX-CLIQUE within if the verifier reads O(q(n)) bits of the debate. a factor of ne is as difficult as solving NP-complete problems exactly [1]. Furthermore, there is a large We then use our main result about the language- class of natural optimization problems, those complete recognition power of PCDS’s to prove lower bounds for the class MAX-SNP defined in [15], that do not on the difficulty of approximating PSPACEhard func- have polynomial-time approximation schemes unless P tions. Let MAX Q3SAT be the following natural opti- = NP; that is, for each of these problems, there is an mization version of the canonical PSPACEcomplete e such that approximating the optimal solution within language QBF. Suppose @ = QIZIQZZZ . Qnxn ratio ~ is as hard as solving NP-complete problems ex- 4(zl, z2,..., zn ) is a quantified boolean formula, with actly [1]. This result on MAX-SNP shows that many Qi c {3, V}, and ~ in 3CNF. Suppose that the vari- well-known optimization problems are hard to approx- ables of the formula are chosen, in order of quantifica- imate closely, including Traveling Salesman with Tri- tion, by two players O and 1, where player O chooses the angle Inequality, MAX-SAT, and MAX-CUT. universally quantified variables and player 1 chooses A PCDS is a generalization of a PCPS. In a PCDS the existentially quantified variables. If player 1 can for L, there are two computationally powerful players, guarantee that k clauses of @ will be satisfied by 1 and O (called B and C at the beginning of this sec- the resulting assignment, regardless of what player O tion) and a probabilistic polynomial-time verifier V. chooses, we say that k clauses of @ are simultaneously Players 1 and O play a game in which they alternate satisfiable. We let MAX Q3SAT be the function that writing out strings on a debate tape iT. Player 1‘s goal maps a quantified 3CNF formula @ to the maximum is to convince V that an input a c L, and player O’s number of simultaneously satisfiable clauses. goal is to convince V that x @ L. When the debate is over, V looks at x and T and decides whether x c L (player 1 wins the debate) or z @ L (player O wins the Theorem: There is a constant O < c <1 such that ap- debate). Suppose V flips O(r(n)) random coins and proximating MAX Q3SAT within ratio c is PSPACE- reads O(q(n)) bits of ~. If, under the best strategies hard. Thus MAX Q3SAT is as hard to approximate of players 1 and O, V’s decision is correct with high as it is to compute exactly. probability, then we say that L is in PCD(r(n), q(n)). Specifically, we say that a language L is in We use reductions to prove that certain other PCD(r(n), q(n)) if it has a nonadaptive PCDS with PSPACEhard functions are PSPACE-hard to approx- one-sided error in which players 1 and O write on a imate in a stronger sense. These include maximiza- debate tape, and then V makes O(r(n)) coin flips and tion versions of the Finite Automata Intersection prob- queries O(q(n)) bits based on these coin flips. By non- lem, shown PSPACEcomplete by Kozen [12], and adaptive, we mean that the choice of bits queried by V the Generalized Geography problem, shown PSPACE- is based solely on the input and the coin flips. By one- complete by Schaefer [17]. We show that there is a con- sided error, we mean that whenever z E L, V must stant ~ such that approximating these problems within correctly decide that x E L, no matter which sequence ratio n’ is PSPACEhard. of O(r(n)) coins are flipped (assuming correct play on the part of player 1). When z # L, V is allowed to The rest of this paper is organized as follows. We conclude incorrectly that z ~ L with probability at define PCDS’S, and all of our other terms, precisely most c, for some fixed ~ < 1. in Section 2. Our results on the language-recognition With the above definition in hand, we can state our power of PCDS’S are given in Section 3. Those on main results about the language-recognition power of approximation of PSPACEhard functions are given PCDS’S. in Section 4. Section 5 contains open questions and a discussion of subsequent related results. Theorem: PSPACE ~ PCD(log n, 1). This re- We have omitted some proofs because of space limi- sult is best possible, because we can show that t ations; they can all be found in our journal submission PCD(log n, q(n)) is contained in PSPACE, for any (DIMACS TR 93-10).