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40
Finding Hard Instances of the Satisfiability Problem: A Survey
, 1997
"... . 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 ..."
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Cited by 114 (1 self)
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. 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.
Typical random 3SAT formulae and the satisfiability threshold
 in Proceedings of the Eleventh ACMSIAM Symposium on Discrete Algorithms
, 2000
"... Abstract: We present a new structural (or syntactic) approach for estimating the satisfiability threshold of random 3SAT formulae. We show its efficiency in obtaining a jump from the previous upper bounds, lowering them to 4.506. The method combines well with other techniques, and also applies to o ..."
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Cited by 87 (2 self)
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Abstract: We present a new structural (or syntactic) approach for estimating the satisfiability threshold of random 3SAT formulae. We show its efficiency in obtaining a jump from the previous upper bounds, lowering them to 4.506. The method combines well with other techniques, and also applies to other problems, such as the 3colourability of random graphs. 1
Approximating the unsatisfiability threshold of random formulas
, 1998
"... ABSTRACT: Let � be a random Boolean formula that is an instance of 3SAT. 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 wellknown and m ..."
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Cited by 82 (14 self)
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ABSTRACT: Let � be a random Boolean formula that is an instance of 3SAT. 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 wellknown 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
Random Constraint Satisfaction: A More Accurate Picture
, 1997
"... 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 "thresholdlike" behaviour and some ..."
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Cited by 77 (7 self)
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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 "thresholdlike" 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
The Probabilistic Analysis of a Greedy Satisfiability Algorithm
, 2002
"... Consider the following simple, greedy DavisPutnam 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 occu ..."
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Cited by 72 (5 self)
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Consider the following simple, greedy DavisPutnam 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.
2+p SAT: Relation of typicalcase complexity to the nature of the phase transition. Random Structures and Algorithms
, 1999
"... ABSTRACT: Heuristic methods for solution of problems in the NPcomplete 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 r ..."
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Cited by 50 (2 self)
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ABSTRACT: Heuristic methods for solution of problems in the NPcomplete 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 KSAT. Studying a model which interpolates KSAT 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 firstorder’ ’ 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
On the solutionspace geometry of random constraint satisfaction problems
 In STOC ’06: Proceedings of the thirtyeighth annual ACM symposium on Theory of computing
, 2006
"... For a large number of random constraint satisfaction problems, such as random kSAT and random graph and hypergraph coloring, there are very good estimates of the largest constraint density for which solutions exist. Yet, all known polynomialtime algorithms for these problems fail to find solutions ..."
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Cited by 45 (2 self)
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For a large number of random constraint satisfaction problems, such as random kSAT and random graph and hypergraph coloring, there are very good estimates of the largest constraint density for which solutions exist. Yet, all known polynomialtime 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 polynomialtime 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
Setting 2 variables at a time yields a new lower bound for random 3SAT (Extended Abstract)
 STOC
, 2000
"... Let X be a set of n Boolean variables and denote by C(X) the set of all 3clauses over X, i.e. the set of all 8(3) possible disjunctions of three distinct, noncomplementary literais from variables in X. Let F(n, m) be a random 3SAT formula formed by selecting, with replacement, m clauses uniformly ..."
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Cited by 35 (4 self)
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Let X be a set of n Boolean variables and denote by C(X) the set of all 3clauses over X, i.e. the set of all 8(3) possible disjunctions of three distinct, noncomplementary literais from variables in X. Let F(n, m) be a random 3SAT formula formed by selecting, with replacement, m clauses uniformly at random from C(X) and taking their conjunction. The satisfiability threshold conjecture asserts that there exists a constant ra such that as n+ c¢, F(n, rn) is satisfiable with probability that tends to 1 if r < ra, but unsatisfiable with probability that tends to 1 if r:> r3. Experimental evidence suggests rz ~ 4.2. We prove rz> 3.145 improving over the previous best lower bound r3> 3.003 due to Frieze and Suen. For this, we introduce a satisfiability heuristic that works iteratively, permanently setting the value of a pair of variables in each round. The framework we develop for the analysis of our heuristic allows us to also derive most previous lower bounds for random 3SAT in a uniform manner and with little effort.
Threshold values of random kSAT from the cavity method. Random Structures and Algorithms (to appear
, 2006
"... Using the cavity equations of [23,24], we derive the various threshold values for the number of clauses per variable of the random Ksatisfiability problem, generalizing the previous results to K ≥ 4. We also give an analytic solution of the equations, and some closed expressions for these threshold ..."
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Cited by 30 (4 self)
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Using the cavity equations of [23,24], we derive the various threshold values for the number of clauses per variable of the random Ksatisfiability problem, generalizing the previous results to K ≥ 4. We also give an analytic solution of the equations, and some closed expressions for these thresholds, in an expansion around large K. The stability of the solution is also computed. For any K, the satisfiability threshold is found to be in the stable region of the solution, which adds further credit to the
Threshold Phenomena in Random Graph Colouring and Satisfiability
, 1999
"... 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 ..."
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Cited by 24 (4 self)
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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 kcolourable]. 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 kcolourability 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