We prove optimal, up to an arbitrary e > 0, inapproximability results for Max-E k-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover.

Some optimal inapproximability results

We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof” with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [1998] whose verifiers examine a nonconstant number of bits in the proof (though this number is a very slowly growing function of the input length).As a consequence, we prove that no MAX SNP-hard problem has a polynomial time approximation scheme, unless NP = P. The class MAX SNP was defined by Papadimitriou and Yannakakis [1991] and hard problems for this class include vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige et al. [1996] and Arora and Safra [1998] and show that there exists a positive e such that approximating the maximum clique size in an N-vertex graph to within a factor of Ne is NP-hard.

Proof verification and the hardness of approximation problems

In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

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Property testing and its connection to learning and approximation

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 $\NP\neq\coRP$, 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.

Free Bits, PCPs, and Nonapproximability---Towards Tight Results

We give an overview of the LEDA platform for combinatorial and geometric computing and an account of its development. We discuss our motivation for building LEDA and to what extent we have reached our goals. We also discuss some recent theoretical developments. This paper contains no new technical material. It is intended as a guide to existing publications about the system. We refer the reader also to our web-pages for more information.

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LubyMichael

Papers

Self-testing/correcting with applications to numerical problems