Let @math be a 3CNF formula with n variables and m clauses. A simple nonconstructive argument shows that when m is sufficiently large compared to n, most 3CNF formulas are not satisfiable. It is an open question whether there is an efficient refutation algorithm that for most such formulas proves that they are not satisfiable. A possible approach to refute a formula @math is: first, translate it into a graph @math using a generic reduction from 3-SAT to max-IS, then bound the maximum independent set of @math using the Lovasz @math function. If the @math function returns a value @math , this is a certificate for the unsatisfiability of @math . We show that for random formulas with @math clauses, the above approach fails, i.e. the @math function is likely to return a value of m.
Further motivation for studying efficient refutability of random 3CNF formulas is given in @cite_1 . There it is shown that if there is no polynomial time refutation heuristic that works for most 3CNF formulas with @math clauses (where @math is an arbitrarily large constant) then certain combinatorial optimization problems (like minimum graph bisection, the dense @math -subgraph, and others) have no polynomial time approximation schemes. It is an open question whether it is NP-hard to approximate these problems arbitrarily well, though further evidence that these problems are indeed hard to approximate is given in @cite_14 .
@cite_14 @cite_1
1983171306 2000931246 99448414 2091602684
We present an algorithm for solving 3SAT instances. Several algorithms have been proved to work whp (with high probability) for various SAT distributions. However, an algorithm that works whp has a drawback. Indeed for typical instances it works well, however for some rare inputs it does not provide a solution at all. Alternatively, one could require that the algorithm always produce a correct answer but perform well on average. Expected polynomial time formalizes this notion. We prove that for some natural distribution on 3CNF formulas, called planted 3SAT, our algorithm has expected polynomial (in fact, almost linear) running time. The planted 3SAT distribution is the set of satisfiable 3CNF formulas generated in the following manner. First, a truth assignment is picked uniformly at random. Then, each clause satisfied by it is included in the formula with probability p. Extending previous work for the planted 3SAT distribution, we present, for the first time for a satisfiable SAT distribution, an expected polynomial time algorithm. Namely, it solves all 3SAT instances, and over the planted distribution (with p = d n2, d > 0 a sufficiently large constant) it runs in expected polynomial time. Our results extend to k-SAT for any constant k. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 Finding a satisfying assignment for a k-CNF formula (k ≥ 3), assuming such exists, is a notoriously hard problem. In this work we consider the uniform distribution over satisfiable k-CNF formulas with a linear number of clauses (clause-variable ratio greater than some constant). We rigorously analyze the structure of the space of satisfying assignments of a random formula in that distribution, showing that basically all satisfying assignments are clustered in one cluster, and agree on all but a small, though constant proportion, number of variables. This observation enables us to describe a polynomial time algorithm that finds whp a satisfying assignment for such formulas, thus asserting that most satisfiable k-CNF formulas are easy (whenever the clause-variable ratio is greater than some constant). This should be contrasted with the setting of very sparse k-CNF formulas (which are satisfiable whp), where experimental results show some regime of clause density to be difficult for many SAT heuristics. One explanation for this phenomena, backed up by partially non-rigorous analytical tools from statistical physics, is the complicated clustering of the solution space at that regime, unlike the more “regular” structure that denser formulas possess. Thus in some sense, our result rigorously supports this explanation. key words: computational and structural complexity, algorithms and data structures, message passing algorithms, SAT. Assuming that NP @math @math BPTIME( @math ), we show that graph min-bisection, dense @math -subgraph, and bipartite clique have no polynomial time approximation scheme (PTAS). We give a reduction from the minimum distance of code (MDC) problem. Starting with an instance of MDC, we build a quasi-random probabilistically checkable proof (PCP) that suffices to prove the desired inapproximability results. In a quasi-random PCP, the query pattern of the verifier looks random in a certain precise sense. Among the several new techniques we introduce, the most interesting one gives a way of certifying that a given polynomial belongs to a given linear subspace of polynomials. As is important for our purpose, the certificate itself happens to be another polynomial, and it can be checked probabilistically by reading a constant number of its values.