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Algorithms for the Satisfiability (SAT) Problem: A Survey
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1996
"... . The satisfiability (SAT) problem is a core problem in mathematical logic and computing theory. In practice, SAT is fundamental in solving many problems in automated reasoning, computeraided design, computeraided manufacturing, machine vision, database, robotics, integrated circuit design, compute ..."
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Cited by 125 (3 self)
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. The satisfiability (SAT) problem is a core problem in mathematical logic and computing theory. In practice, SAT is fundamental in solving many problems in automated reasoning, computeraided design, computeraided manufacturing, machine vision, database, robotics, integrated circuit design, computer architecture design, and computer network design. Traditional methods treat SAT as a discrete, constrained decision problem. In recent years, many optimization methods, parallel algorithms, and practical techniques have been developed for solving SAT. In this survey, we present a general framework (an algorithm space) that integrates existing SAT algorithms into a unified perspective. We describe sequential and parallel SAT algorithms including variable splitting, resolution, local search, global optimization, mathematical programming, and practical SAT algorithms. We give performance evaluation of some existing SAT algorithms. Finally, we provide a set of practical applications of the sat...
Generating Hard Satisfiability Problems
 Artificial Intelligence
, 1996
"... We report results from largescale experiments in satisfiability testing. As has been observed by others, testing the satisfiability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible ..."
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Cited by 98 (2 self)
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We report results from largescale experiments in satisfiability testing. As has been observed by others, testing the satisfiability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible to generate random formulas that are hard, that is, for which satisfiability testing is quite difficult. Our results provide a benchmark for the evaluation of satisfiabilitytesting procedures. In Artificial Intelligence, 81 (19996) 1729. 1 Introduction Many computational tasks of interest to AI, to the extent that they can be precisely characterized at all, can be shown to be NPhard in their most general form. However, there is fundamental disagreement, at least within the AI community, about the implications of this. It is claimed on the one hand that since the performance of algorithms designed to solve NPhard tasks degrades rapidly with small increases in input size, something ...
The scaling window of the 2sat transition
, 1999
"... Abstract. We consider the random 2satisfiability problem, in which each instance is a formula that is the conjunction of m clauses of the form x ∨ y, chosen uniformly at random from among all 2clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/n → α, the ..."
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Cited by 24 (1 self)
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Abstract. We consider the random 2satisfiability problem, in which each instance is a formula that is the conjunction of m clauses of the form x ∨ y, chosen uniformly at random from among all 2clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/n → α, the problem is known to have a phase transition at αc = 1, below which the probability that the formula is satisfiable tends to one and above which it tends to zero. We determine the finitesize scaling about this transition, namely the scaling of the maximal window W(n,δ) = (α−(n,δ),α+(n,δ)) such that the probability of satisfiability is greater than 1 − δ for α < α − and is less than δ for α> α+. We show that W(n,δ) = (1 − Θ(n −1/3),1 + Θ(n −1/3)), where the constants implicit in Θ depend on δ. We also determine the rates at which the probability of satisfiability approaches one and zero at the boundaries of the window. Namely, for m = (1 + ε)n, where ε may depend on n as long as ε  is sufficiently small and εn 1/3 is sufficiently large, we show that the probability of satisfiability decays like exp ( −Θ ( nε 3)) above the window, and goes to one like 1 − Θ ( n −1 ε  −3) below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in n both inside and outside the window. Using this order parameter, we prove that the 2SAT phase transition is continuous with an order parameter critical exponent of 1. We also determine the values of two other critical exponents, showing that the exponents of 2SAT are identical to those of the random graph.
Global Optimization for Satisfiability (SAT) Problem
, 1994
"... The satisfiability (SAT) problem is a fundamental problem in mathematical logic, inference, automated reasoning, VLSI engineering, and computing theory. In this paper, following CNF and DNF local search methods, we introduce the Universal SAT problem model, UniSAT, that transforms the discrete SAT ..."
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Cited by 20 (3 self)
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The satisfiability (SAT) problem is a fundamental problem in mathematical logic, inference, automated reasoning, VLSI engineering, and computing theory. In this paper, following CNF and DNF local search methods, we introduce the Universal SAT problem model, UniSAT, that transforms the discrete SAT problem on Boolean space f0; 1g m into an unconstrained global optimization problem on real space E m . A direct correspondence between the solution of the SAT problem and the global minimum point of the UniSAT objective function is established. Many existing global optimization algorithms can be used to solve the UniSAT problems. Combined with backtracking /resolution procedures, a global optimization algorithm is able to verify satisfiability as well as unsatisfiability. This approach achieves significant performance improvements for certain classes of conjunctive normal form (CNF ) formulas. It offers a complementary approach to the existing SAT algorithms.
Algorithms for counting 2SAT solutions and colorings with applications
 TR05033, Electronic Colloquium on Computational Complexity
, 2005
"... An algorithm is presented for exactly solving (in fact, counting) the number of maximum weight satisfying assignments of a 2Cnf formula. The worst case running time of ..."
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Cited by 18 (2 self)
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An algorithm is presented for exactly solving (in fact, counting) the number of maximum weight satisfying assignments of a 2Cnf formula. The worst case running time of
A Tighter Bound for Counting MaxWeight Solutions to 2SAT Instances
"... We give an algorithm for counting the number of maxweight solutions to a 2SAT formula, and improve the bound on its running time to O (1.2377 n). The main source of the improvement is a refinement of the method of analysis, where we extend the concept of compound (piecewise linear) measures to mult ..."
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Cited by 7 (0 self)
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We give an algorithm for counting the number of maxweight solutions to a 2SAT formula, and improve the bound on its running time to O (1.2377 n). The main source of the improvement is a refinement of the method of analysis, where we extend the concept of compound (piecewise linear) measures to multivariate measures, also allowing the optimal parameters for the measure to be found automatically. This method extension should be of independent interest.
Approximate Resolution of Hard Numbering Problems
, 1996
"... We present a new method for estimating the number of solutions of constraint satisfaction problems 1 . We use a stochastic forward checking algorithm for drawing a sample of paths from a search tree. With this sample, we compute two values related to the number of solutions of a CSP instance. ..."
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Cited by 6 (2 self)
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We present a new method for estimating the number of solutions of constraint satisfaction problems 1 . We use a stochastic forward checking algorithm for drawing a sample of paths from a search tree. With this sample, we compute two values related to the number of solutions of a CSP instance. First, an unbiased estimate, second, a lower bound with an arbitrary low error probability. We will describe applications to the Boolean Satisfiability problem and the Queens problem. We shall give some experimental results for these problems. Introduction The class NP is the set of decision problems whose instances are assertions that can be proved in polynomial time. The NPComplete problems are the hardest problems in NP. These can not be solved in polynomial time under the assumption P 6= NP . All these problems have the same expression power in the sense that every NPComplete problem can be polynomialy reduced to each another (Garey & Johnson 1979). Some of them, such as CSP 2...
Counting Models for 2SAT and 3SAT Formulae
"... We here present algorithms for counting models and maxweight models for 2sat and 3sat formulae. They use polynomial space and run in O(1.2561^n) and O(1.6737^n) time, respectively, where n is the number of variables. This is faster than the previously best algorithms for counting nonweighted model ..."
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Cited by 5 (0 self)
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We here present algorithms for counting models and maxweight models for 2sat and 3sat formulae. They use polynomial space and run in O(1.2561^n) and O(1.6737^n) time, respectively, where n is the number of variables. This is faster than the previously best algorithms for counting nonweighted models for 2sat and 3sat, which run in O(1.3247^n) and O(1.6894^n) time, respectively. In order to prove these time bounds, we develop new measures of formula complexity, allowing us to conveniently analyze the eoeects of certain factors with a large impact on the total running time. We also provide an algorithm for the restricted case of separable 2sat formulae, with fast running times for wellstudied input classes. For all three algorithms we present interesting applications, such as computing the permanent of sparse 0/1 matrices.
Faithful Representations and Moments of Satisfaction: Probabilistic Methods in Learning and Logic
, 1998
"... ii To my wife, Ma'ayan, and my daughter, Shira. iii Acknowledgments Special thanks are due to: ffl Prof. Naftali Tishby for his help and guidance in carrying out this study, for the many fascinating discussions we had, and for the immense body of knowledge that I have absorbed from him during my stu ..."
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Cited by 2 (0 self)
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ii To my wife, Ma'ayan, and my daughter, Shira. iii Acknowledgments Special thanks are due to: ffl Prof. Naftali Tishby for his help and guidance in carrying out this study, for the many fascinating discussions we had, and for the immense body of knowledge that I have absorbed from him during my studies.
New upper bound for the #3SAT problem
"... We present a new deterministic algorithm for the #3SAT problem, based on the DPLL strategy. It uses a new approach for counting models of instances with low density. This allows us to assume the adding of more 2clauses than in previous algorithms. The algorithm achieves a running time of O(1.6423 ..."
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Cited by 2 (0 self)
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We present a new deterministic algorithm for the #3SAT problem, based on the DPLL strategy. It uses a new approach for counting models of instances with low density. This allows us to assume the adding of more 2clauses than in previous algorithms. The algorithm achieves a running time of O(1.6423 n) in the worst case which improves the current best bound of O(1.6737 n) by Dahllöf et al. 1