. Finding sets of hard instances of propositional satisfiability is of interest for understanding the complexity of SAT, and for experimentally evaluating SAT algorithms. In discussing this we consider the performance of the most popular SAT algorithms on random problems, the theory of average case complexity, the threshold phenomenon, known lower bounds for certain classes of algorithms, and the problem of generating hard instances with solutions.

ABSTRACT: Let � be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number � such that if the ratio of the number of clauses over the number of variables of � strictly exceeds �, then � is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that ��5.191. This upper bound was improved by Kamath et al. to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of � is around 4.2. In this work, we define, in terms of the random formula �, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then � 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 � equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601�. In general, by letting the

Recently there has been a great amount of interest in Random Constraint Satisfaction Problems, both from an experimental and a theoretical point of view. Rather intruigingly, experimental results with various models for generating random CSP instances suggest a "threshold-like" behaviour and some theoretical work has been done in analyzing these models when the number of variables is asymptotic. In this paper we show that the models commonly used for generating random CSP instances suffer from a wrong parameterization which makes them unsuitable for asymptotic analysis. In particular, when the number of variables becomes large almost all instances they generate are, trivially, overconstrained. We then present a new model that is suitable for asymptotic analysis and, in the spirit of random SAT, we derive lower and upper bounds for its parameters so that the instances generated are "almost surely" over- and underconstrained, respectively. Finally, we apply the technique introduced in [19], to one of the popular models in Artificial Intelligence and derive sharper estimates for the probability of being overconstrained as a function of the number of variables. 1

this paper, in order to estimate ! better, a new `syntactic' approach is proposed: This is suggested, e.g., simply by practical experience with a computer-based random generator and solver of formulae. The generator is completely ignorant of semantics, yet for large n, it produces (within a realistic timeframe) either only satisfiable formulae, for values of c a little below 4:25, or only unsatisfiable formulae, for values of c a little above 4:25. Only a certain class of formulae is likely to come out of the generator, the `typical' formulae. If we could describe (to a sufficient degree) their syntactic structure, the convergence to 0 or to +1 of the expectation of the number of solutions, computed only for these typical formulae, should give a very good indication as to the value of the threshold.

Consider the following simple, greedy Davis-Putnam algorithm applied to a random 3CNF formula of fixed density (clauses to variables ratio): Arbitrarily select and set to True a literal that appears in as many clauses as possible, irrespective of their size (and irrespective of the number of occurrences of the negation of the literal). Delete these clauses from the formula, and also delete the negation of this literal from any clauses it appears. Repeat. If however unit clauses ever appear, then first repeatedly and in any order set the literals in them to True and delete and shrink clauses accordingly, until no unit clause remains. Also if at any step an empty clause appears, then do not backtrack, but just terminate the algorithm and report failure. A slight modification of this algorithm is probabilistically analyzed in this paper (rigorously). It is proved that for random formulas of n variables and density up to 3.42, it succeeds in producing a satisfying truth assignment with bounded away from zero probability, as n approaches infinity. Therefore the satisfiability threshold is at least 3.42.

ABSTRACT: Heuristic methods for solution of problems in the NP-complete class of decision problems often reach exact solutions, but fail badly at ‘‘phase boundaries,’ ’ across which the decision to be reached changes from almost always having one value to almost always having a different value. We report an analytic solution and experimental investigations of the phase transition that occurs in the limit of very large problems in K-SAT. Studying a model which interpolates K-SAT between K�2 and K�3, we find a change from a continuous to a discontinuous phase transition when K, the average number of inputs per clause, exceeds 0.4. The cost of finding solutions also increases dramatically above this changeover. The nature of its ‘‘random first-order’ ’ phase transition, seen at values of K large enough to make the computational cost of solving typical instances increase exponentially with problem size, suggests a mechanism for the cost increase. There has been

For a large number of random constraint satisfaction problems, such as random k-SAT and random graph and hypergraph coloring, there are very good estimates of the largest constraint density for which solutions exist. Yet, all known polynomial-time algorithms for these problems fail to find solutions even at much lower densities. To understand the origin of this gap we study how the structure of the space of solutions evolves in such problems as constraints are added. In particular, we prove that much before solutions disappear, they organize into an exponential number of clusters, each of which is relatively small and far apart from all other clusters. Moreover, inside each cluster most variables are frozen, i.e., take only one value. The existence of such frozen variables gives a satisfying intuitive explanation for the failure of the polynomial-time algorithms analyzed so far. At the same time, our results establish rigorously one of the two main hypotheses underlying Survey Propagation, a heuristic introduced by physicists in recent years that appears to perform extraordinarily well on random constraint satisfaction problems. 1

We study threshold phenomena pertaining to the colourability of random graphs and the satisfiability of random formulas. Consider a random graph G(n, p) on n vertices formed by including each of the possible edges independently of all others with probability p. For a fixed integer k, let f k (n, d) = Pr[G(n, d/n) is k-colourable]. Erdos asked the following fundamental question: for k 3, is there a constant c k such that for any # > 0, #) = 1 , and lim f k (n, c k + #) = 0 ? (1) We prove that for all k 3, there exists a function t k (n) such that (1) holds upon replacing c k by t k (n), thus establishing that indeed k-colourability has a sharp threshold. Let d k = sup{d lim n## f k (n, d) = 1}. Note that if c k exists then, by definition, c k = d k . For the basic and most studied case k = 3 we prove 3.84 < d 3 < 5.05 . These are the best

ix Chapter 1 Introduction 1 1.1 Algorithmic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Parallel Computational Complexity . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.5 Random Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.6 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 2 Parallel Uniform Generation of Unlabelled Graphs 25 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Sampling O...

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