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Efficient Probabilistically Checkable Proofs and Applications to Approximation Abstract 1 Introduction Appears in Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, ACM (1993). Efficient Probabilistically Checkable Proofs and Applications to Approximation M. Bellare∗ S. Goldwassery C. Lundz A. Russellx May 1993 Abstract 1 Introduction We construct multi-prover proof systems for NP which The last two years have witnessed major advances in use only a constant number of provers to simultaneously classifying the complexity of approximating classical op- achieve low error, low randomness and low answer size. timization problems [Co, FGLSS, AS, ALMSS, BR, FL, As a consequence, we obtain asymptotic improvements Be, Zu, LY1, LY2]. These works indicate that a prob- to approximation hardness results for a wide range of lem P is hard to approximate by showing that the ex- optimization problems including minimum set cover, istence of a polynomial time algorithm that approxi- dominating set, maximum clique, chromatic number, mates problem P to within some factor Q would im- and quartic programming; and constant factor improve- ply an unlikely conclusion like NP ⊆ DTIME(T (n)) ments on the hardness results for MAXSNP problems. or NP ⊆ RTIME(T (n)), with T polynomial or quasi- In particular, we show that approximating minimum polynomial. These results are derived by reductions set cover within any constant is NP-complete; approx- from interactive proofs. Namely, by first characteriz- imating minimum set cover within Θ(log n) implies ing NP as those languages which have efficient interac- NP ⊆ DTIME(nlog log n); approximating the maximum tive proofs of membership; and second by reducing the of a quartic program within any constant is NP-hard; problem of whether there exists an interactive proof for approximating maximum clique within n1=30 implies membership in L (L 2 NP) to the problem of approxi- NP ⊆ BPP; approximating chromatic number within mating the cost of an optimal solution to an instance of n1=146 implies NP ⊆ BPP; and approximating MAX- problem P . 3SAT within 113=112 is NP-complete. Today such results are known for many important ∗Department of Computer Science & Engineering, Mail Code problems P , with values of Q and T which differ from 0114, University of California at San Diego, 9500 Gilman Drive, problem to problem; for example, it was shown by [LY1] La Jolla, CA 92093. E-mail: [email protected]. y that approximating the size of the minimum set cover to MIT Laboratory for Computer Science, 545 Technology polylog n Square, Cambridge, MA 02139, USA. e-mail: shafi@theory. within Θ(log N) implies NP ⊆ DTIME(n ), and lcs.mit.edu. Partially supported by NSF FAW grant No. it was shown by [FGLSS, AS, ALMSS] that for some 9023312-CCR, DARPA grant No. N00014-92-J-1799, and grant constant c > 0 approximating the size of a maximum No. 89-00312 from the United States - Israel Binational Science clique in a graph within factor nc implies that P = NP. Foundation (BSF), Jerusalem, Israel. zAT&T Bell Laboratories, Room 2C324, 600 Mountain Av- The values of Q and T achieved depend on the effi- enue, P. O. Box 636, Murray Hill, NJ 07974-0636, USA. email: ciency parameters of the underlying proof system used [email protected]. in the reduction to optimization problem P . The precise x MIT Laboratory for Computer Science, 545 Technology manner in which Q and T depend on these parameters Square, Cambridge, MA 02139, USA. e-mail: [email protected]. mit.edu. Supported by a NSF Graduate Fellowship and by NSF depends on the particular problem P and the particular grant 92-12184, AFOSR 89-0271, and DARPA N00014-92-J-1799. reduction used. 1 r = r(n) p = p(n) a = a(n) = (n) How (in a word) 1 (1) O(log n) 2 O(1) 2 + ∗ [ALMSS]+[FRS]+[Fe] (2) O(log n) O(k(n)) O(1) 2−k(n) O(k(n)) [CW, IZ]-style repetitions of (1). (3) O(k(n) log2 n) 2 O(k(n) log2 n) 2−k(n) [FL] (4) O(k(n) log n) + poly(k(n); log log n) 4 poly(k(n); log log n) 2−k(n) This paper. Figure 1: Number of random bits (r), number of provers (p), answer size (a) and error probability () in results of the form NP ⊆ MIP1[r; p; a; q; ]. Here k(n) is any function bounded above by O(log n) and ∗ is any positive constant. The goal of this paper is to improve the values of We emphasize that the model (PCP or MIP1) is not Q and T in such reductions. Thus we need to reduce as important, in this context, as the values of the pa- the complexity of the underlying proof systems. Let us rameters p; r; a; . Although the parameterized versions begin by seeing what are the proof systems in question of PCP and MIP1 are not known to have equal language and what within these proof systems are the complexity recognition power for a given r; p; a; ,1 most known re- parameters we need to consider. ductions to approximation problems in one model are easily modified to work in the other as long as p; r; a; remain the same. Accordingly, we sometimes state re- 1.1 PCP and MIP sults in terms of MIP1 and sometimes in terms of PCP, Several variants of the interactive proof model have been and leave the translations to the reader. used to derive intractability of approximation results. We note that there may be a motivation to move to We focus on two of them. The first is the (single round the PCP model when proving an approximation hard- version of the) multi-prover model of Ben-Or, Gold- ness result if one could prove results of the form NP ⊆ wasser, Kilian and Wigderson [BGKW]. The second PCP[r; p; a; q; ] which attain better values of the param- is the \oracle" model of Fortnow, Rompel and Sipser eters than results of the form NP ⊆ MIP1[r; p; a; q; ]. [FRS], renamed \probabilistically checkable proofs" by Arora and Safra [AS]. In each case, we may distin- 1.2 New Proof Systems for NP guish five parameters which we denote by r; p; a; q and (all are in general functions of the input length n). In a Our main result is the construction of low complexity, multi prover proof these are, respectively, the number of low error proof systems for NP which have only a con- random bits used by the verifier, the number of provers, stant number of provers. In its most general form the the size of each prover's answer, the size of each of the result is the following. verifier’s questions to the provers, and the error proba- bility. Correspondingly in a probabilistically checkable Theorem 1.1 Let k(n) ≤ O(log n) be any func- −k(n) proof these are the number of random bits used by the tion. Then NP ⊆ MIP1[r; 4; a; q; 2 ], where verifier, the number of queries to the oracle, the size r = O(k(n) log n) + poly(k(n); log log n), a = of each of the oracle's answers, the size of each indi- poly(k(n); log log n), and q = O(r). vidual query, and the error probability. We denote by MIP1[r; p; a; q; ] and PCP[r; p; a; q; ] the corresponding The table of Figure 1 summarizes previous results of classes of languages. the form NP ⊆ MIP1[r; p; a; q; ] in comparison with Note that the total number of bits returned by the ours. We omit the question size q from the table be- provers or oracle is pa; we will sometimes denote this cause it is in all cases O(r) and doesn't matter in re- quantity by t. In some applications the important pa- ductions anyway. The main result of [ALMSS], which rameter is the expected value of t. To capture this we states that there is a constant t such that NP = 0 define PCP as PCP except that the p parameter is the PCP[O(log n); t; 1;O(log n); 1=2], is incorporated as the expected number of queries made by the verifier. 1 The parameters which are important in applications It is easy to see that MIP1[r; p; a; q; ] ⊆ PCP[r; p; a; q; ], to approximation seem to be r; p; a; . However, as q is but the converse containment is not known to be true. When complexity is ignored the models are of course the same [FRS, important in transformations of one type of proof sys- BFL]. For explanations and more information we refer the reader tem to another, we included it in the list. to §2. 2 special case k(n) = 1 of the result shown in (2). The re- and quartic programming (using Theorem 1.1), and we sult shown in (1) is obtained as follows. First apply the have improvements in the factor Q for maximum clique, transformation of [FRS] to the [ALMSS] result to get chromatic number, MAX3SAT, and quadratic program- NP ⊆ MIP1[O(log n); 2; t; O(log n); 1 − 1=(2t)] where t ming (using Theorem 1.2). is the constant from [ALMSS] as above. Then apply [Fe] to bring the error to any constant strictly greater than Set Cover. For a definition of the problem we re- 1=2, at constant factor cost in the other parameters. fer the reader to §4.1. Recall that there exists a poly- Comparing these results with ours, we note the follow- nomial time algorithm for approximating the size of ing features, all of which are important to our applica- the minimum set cover to within a factor of Θ(log N), tions.
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