MAX-3SAT

Abstract: We give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof.We discuss implications of this characterization; specifically, we show that approximating Clique and Independent Set, even in a very weak sense, is NP-hard.

Abstract: The class PCP(f(n),g(n)) consists of all languages L for which there exists a polynomial-time probabilistic oracle machine that used O(f(n)) random bits, queries O(g(n)) bits of its oracle and behaves as follows: If x in L then there exists an oracle y such that the machine accepts for all random choices but if x not in L then for every oracle y the machine rejects with high probability. Arora and Safra (1992) characterized NP as PCP(log n, (loglogn)/sup O(1)/). The authors improve on their result by showing that NP=PCP(logn, 1). The result has the following consequences: (1) MAXSNP-hard problems (e.g. metric TSP, MAX-SAT, MAX-CUT) do not have polynomial time approximation schemes unless P=NP; and (2) for some epsilon >0 the size of the maximal clique in a graph cannot be approximated within a factor of n/sup epsilon / unless P=NP. >

Abstract: This paper continues the investigation of the connection between probabilistically checkable proofs (PCPs) and the approximability of NP-optimization problems. The emphasis is on proving tight nonapproximability results via consideration of measures such as the "free-bit complexity" and the "amortized free-bit complexity" of proof systems. The first part of the paper presents a collection of new proof systems based on a new error-correcting code called the long code. We provide a proof system that has amortized free-bit complexity of $2 + \epsilon$, implying that approximating MaxClique within $N^{\frac13-\e}$, and approximating the Chromatic Number within $N^{\frac15-\e}$, are hard, assuming $\NP eq\coRP$, for any e > 0. We also derive the first explicit and reasonable constant hardness factors for Min Vertex Cover, $\MSAT{2}$, and Max Cut, and we improve the hardness factor for $\MSAT{3}$. We note that our nonapproximability factors for $\maxsnp$ problems are appreciably close to the values known to be achievable by polynomial-time algorithms. Finally, we note a general approach to the derivation of strong nonapproximability results under which the problem reduces to the construction of certain "gadgets." The increasing strength of nonapproximability results obtained via the PCP connection motivates us to ask how far this can go and whether PCPs are inherent in any way. The second part of the paper addresses this. The main result is a "reversal" of the connection due to Feige et al. (FGLSS connection) [J. ACM, 43 (1996), pp. 268--292]: where the latter had shown how to translate proof systems for NP into NP-hardness of approximation results for MaxClique, we show how any NP-hardness of approximation result for MaxClique yields a proof system for NP. Roughly, our result says that for any constant f, if MaxClique is NP-hard to approximate within N 1(1+f), then $\NP\subseteq \overline{\fpcp}[\log,f]$, the latter being the class of languages possessing proofs of logarithmic randomness and amortized free-bit complexity f. This suggests that PCPs are inherent to obtaining nonapproximability results. Furthermore, the tight relation suggests that reducing the amortized free-bit complexity is necessary for improving the nonapproximability results for MaxClique. The third part of our paper initiates a systematic investigation of the properties of PCP and FPCP (free PCP) as a function of the following various parameters: randomness, query complexity, free-bit complexity, amortized free-bit complexity, proof size, etc. We are particularly interested in triviality results, which indicate which classes are not powerful enough to capture NP. We also distill the role of randomized reductions in this area and provide a variety of useful transformations between proof checking complexity classes.