Journal Article•
The probabilistic analysis of a greedy satisfiability algorithm
TL;DR: In this article, a simple greedy Davis-Putnam algorithm is applied to a random 3-CNF formula of constant density c: Arbitrarily 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.
Abstract: Consider the following simple, greedy Davis-Putnam algorithm applied to a random 3-CNF formula of constant density c: Arbitrarily 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). Reduce the formula. If any unit clauses appear, then satisfy their literals arbitrarily, reducing the formula accordingly, until no unit clause remains. Repeat. We prove that for c < 3.42 a slight modification of this algorithm computes a satisfying truth assignment with probability asymptotically bounded away from zero. Previously, algorithms of increasing sophistication were shown to succeed for c < 3.26. Preliminary experiments we performed suggest that c ≃ 3.6 is feasible running algorithms like the above, which take into account not only the number of occurrences of a literal but also the number of occurrences of its negation, irrespectively of clause-size information.
Citations
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01 Jan 2004
TL;DR: A case study for Efficient Implementation of Algorithms of SAT Solvers and the Interaction Between Inference and Branching Heuristics.
Abstract: ion-Driven SAT-based Analysis of Security Protocols . . . . . . . . . . 257 Alessandro Armando, Luca Compagna A Case for Efficient Solution Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Sarfraz Khurshid, Darko Marinov, Ilya Shlyakhter, Daniel Jackson Cache Performance of SAT Solvers: a Case Study for Efficient Implementation of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Lintao Zhang, Sharad Malik Local Consistencies in SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Christian Bessière, Emmanuel Hebrard, Toby Walsh Guiding SAT Diagnosis with Tree Decompositions . . . . . . . . . . . . . . . . . . . . 315 Per Bjesse, James Kukula, Robert Damiano, Ted Stanion, Yunshan Zhu On Computing k-CNF Formula Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 Ryan Williams Effective Preprocessing with Hyper-Resolution and Equality Reduction . . 341 Fahiem Bacchus, Jonathan Winter Read-Once Unit Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Hans Kleine Büning, Xishun Zhao The Interaction Between Inference and Branching Heuristics . . . . . . . . . . . . 370 Lyndon Drake, Alan Frisch Hypergraph Reductions and Satisfiability Problems . . . . . . . . . . . . . . . . . . . 383 Daniele Pretolani Table of
137 citations
01 Jan 2006
TL;DR: The goal of this work is to review the series of improvements of the upper bounds for 3-SAT and the techniques from which the improvements resulted, as they rely on significantly different techniques that would require much more space to present.
Abstract: One of the most challenging problems in probability and complexity theory concerns the establishment and the determination of the satisfiability threshold for random Boolean formulas consisting of clauses with exactly k literals, or k-SAT formulas with emphasis on the case k = 3, or 3-SAT. According to many experimental observations, there exists a critical value rk of the number of clauses to the number of variables ratio r = m/n such that almost all randomly generated formulas with r > rk are unsatisfiable while almost all randomly generated formulas with r < rk are satisfiable. The statement that such a crossover point really exists is called the “satisfiability threshold conjecture”. While experiments hint at such a direction, as far as theoretical work is concerned, progress has been difficult. Up to now, there are rigorous proofs of only successively better upper and lower bounds for the value of the (conjectured) threshold although, in an important advance, Friedgut proved that the phase transition is sharp (without showing the existence of a fixed transition point). In this work, our goal is to review the series of improvements of the upper bounds for 3-SAT and the techniques from which the improvements resulted. We give only a passing reference to the improvements of the lower bounds, as they rely on significantly different techniques that would require much more space to present.
4 citations
01 Jul 2018
TL;DR: The main result is that the threshold of random community-structured SAT tends to be smaller than its counterpart for random SAT, and under some conditions, this threshold even vanishes.
Abstract: For both historical and practical reasons, the Boolean satisfiability
problem (SAT) has become one of central importance in computer science.
One type of instances arises when the clauses are chosen uniformly
randomly \textendash{} random SAT. Here, a major problem, recently
solved for sufficiently large clause length, is the satisfiability
threshold conjecture. The value of this threshold is known exactly
only for clause length $2$, and there has been a lot of research
concerning its value for arbitrary fixed clause length.
In this paper, we endeavor to study the satisfiability threshold for random industrial SAT. There is as yet no generally accepted model of industrial SAT, and we confine ourselves to one of the more common features of industrial SAT: the set of variables consists of a number of disjoint communities,
and clauses tend to consist of variables from the same community.
Our main result is that the threshold of random community-structured SAT tends
to be smaller than its counterpart for random SAT. Moreover, under
some conditions, this threshold even vanishes.
1 citations
TL;DR: In this article , the authors study the satisfiability threshold for the proposed model of random industrial SAT and show that the threshold tends to be smaller than its counterpart for random SAT, and under some conditions, this threshold even vanishes.
Abstract: One of the most studied models of SAT is random SAT. In this model, instances are composed from clauses chosen uniformly randomly and independently of each other. This model may be unsatisfactory in that it fails to describe various features of SAT instances, arising in real-world applications. Various modifications have been suggested to define models of industrial SAT. Here, we focus mainly on the aspect of community structure. Namely, here the set of variables consists of a number of disjoint communities, and clauses tend to consist of variables from the same community. Thus, we suggest a model of random industrial SAT, in which the central generalization with respect to random SAT is the additional community structure. There has been a lot of work on the satisfiability threshold of random k-SAT, starting with the calculation of the threshold of 2-SAT, up to the recent result that the threshold exists for sufficiently large k. In this paper, we endeavor to study the satisfiability threshold for the proposed model of random industrial SAT. Our main result is that the threshold in this model tends to be smaller than its counterpart for random SAT. Moreover, under some conditions, this threshold even vanishes.
TL;DR: In this paper , a connected treewidth model of the propositional formula was constructed by using the connected tree decomposition method, and the connected trewidth of the factor graph was calculated.
Abstract: The survey propagation algorithm is the most effective information propagation algorithm for solving the 3-SAT problem. It can effectively solve the satisfiability problem when it converges. However, when the factor graph structure is complex, the algorithm often does not converge and the solution fails. In order to give a theoretical explanation to this phenomenon and to analyze the convergence of the survey propagation algorithm effectively, a connected treewidth model of the propositional formula was constructed by using the connected tree decomposition method, and the connected treewidth of the factor graph was calculated. The relationship between the connected treewidth and the convergence of the survey propagation algorithm is established, and the convergence judgment condition of the survey propagation algorithm based on the connected tree width is given. Through experimental analysis, the results show that the method is effective, which is of great significance for analyzing the convergence analysis of other information propagation algorithms.
References
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Book•
31 Aug 2011TL;DR: The programming of a proof procedure is discussed in connection with trial runs and possible improvements.
Abstract: The programming of a proof procedure is discussed in connection with trial runs and possible improvements.
3,296 citations
TL;DR: In the present paper, a uniform proof procedure for quantification theory is given which is feasible for use with some rather complicated formulas and which does not ordinarily lead to exponentiation.
Abstract: The hope that mathematical methods employed in the investigation of formal logic would lead to purely computational methods for obtaining mathematical theorems goes back to Leibniz and has been revived by Peano around the turn of the century and by Hilbert's school in the 1920's. Hilbert, noting that all of classical mathematics could be formalized within quantification theory, declared that the problem of finding an algorithm for determining whether or not a given formula of quantification theory is valid was the central problem of mathematical logic. And indeed, at one time it seemed as if investigations of this “decision” problem were on the verge of success. However, it was shown by Church and by Turing that such an algorithm can not exist. This result led to considerable pessimism regarding the possibility of using modern digital computers in deciding significant mathematical questions. However, recently there has been a revival of interest in the whole question. Specifically, it has been realized that while no decision procedure exists for quantification theory there are many proof procedures available—that is, uniform procedures which will ultimately locate a proof for any formula of quantification theory which is valid but which will usually involve seeking “forever” in the case of a formula which is not valid—and that some of these proof procedures could well turn out to be feasible for use with modern computing machinery.Hao Wang [9] and P. C. Gilmore [3] have each produced working programs which employ proof procedures in quantification theory. Gilmore's program employs a form of a basic theorem of mathematical logic due to Herbrand, and Wang's makes use of a formulation of quantification theory related to those studied by Gentzen. However, both programs encounter decisive difficulties with any but the simplest formulas of quantification theory, in connection with methods of doing propositional calculus. Wang's program, because of its use of Gentzen-like methods, involves exponentiation on the total number of truth-functional connectives, whereas Gilmore's program, using normal forms, involves exponentiation on the number of clauses present. Both methods are superior in many cases to truth table methods which involve exponentiation on the total number of variables present, and represent important initial contributions, but both run into difficulty with some fairly simple examples.In the present paper, a uniform proof procedure for quantification theory is given which is feasible for use with some rather complicated formulas and which does not ordinarily lead to exponentiation. The superiority of the present procedure over those previously available is indicated in part by the fact that a formula on which Gilmore's routine for the IBM 704 causes the machine to computer for 21 minutes without obtaining a result was worked successfully by hand computation using the present method in 30 minutes. Cf. §6, below.It should be mentioned that, before it can be hoped to employ proof procedures for quantification theory in obtaining proofs of theorems belonging to “genuine” mathematics, finite axiomatizations, which are “short,” must be obtained for various branches of mathematics. This last question will not be pursued further here; cf., however, Davis and Putnam [2], where one solution to this problem is given for ele
2,743 citations
TL;DR: For the problem of finding the maximum clique in a graph, no algorithm has been found for which the ratio does not grow at least as fast as n^@e, where n is the problem size and @e>0 depends on the algorithm.
2,472 citations
TL;DR: An exact method is given which performs better than the Randall-Brown algorithm and is able to color larger graphs and the new heuristic methods, the classical methods, and the exact method are compared.
Abstract: This paper describes efficient new heuristic methods to color the vertices of a graph which rely upon the comparison of the degrees and structure of a graph. A method is developed which is exact for bipartite graphs and is an important part of heuristic procedures to find maximal cliques in general graphs. Finally an exact method is given which performs better than the Randall-Brown algorithm and is able to color larger graphs, and the new heuristic methods, the classical methods, and the exact method are compared.
1,510 citations