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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 145 (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...
Tail Bounds for Occupancy and the Satisfiability Threshold Conjecture
, 1995
"... The classical occupancy problem is concerned with studying the number of empty bins resulting from a random allocation of m balls to n bins. We provide a series of tail bounds on the distribution of the number of empty bins. These tail bounds should find application in randomized algorithms and prob ..."
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Cited by 111 (2 self)
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The classical occupancy problem is concerned with studying the number of empty bins resulting from a random allocation of m balls to n bins. We provide a series of tail bounds on the distribution of the number of empty bins. These tail bounds should find application in randomized algorithms and probabilistic analysis. Our motivating application is the following wellknown conjecture on threshold phenomenon for the satisfiability problem. Consider random 3SAT formulas with cn clauses over n variables, where each clause is chosen uniformly and independently from the space of all clauses of size 3. It has been conjectured that there is a sharp threshold for satisfiability at c ß 4:2. We provide a strong upper bound on the value of c , showing that for c ? 4:758 a random 3SAT formula is unsatisfiable with high probability. This result is based on a structural property, possibly of independent interest, whose proof needs several applications of the occupancy tail bounds. Supporte...
Simplified and Improved Resolution Lower Bounds
 IN PROCEEDINGS OF THE 37TH IEEE FOCS
, 1996
"... We give simple new lower bounds on the lengths of Resolution proofs for the pigeonhole principle and for randomly generated formulas. For random formulas, our bounds significantly extend the range of formula sizes for which nontrivial lower bounds are known. For example, we show that with probabili ..."
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Cited by 103 (8 self)
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We give simple new lower bounds on the lengths of Resolution proofs for the pigeonhole principle and for randomly generated formulas. For random formulas, our bounds significantly extend the range of formula sizes for which nontrivial lower bounds are known. For example, we show that with probability approaching 1, any Resolution refutation of a randomly chosen 3CNF formula with at most n 6=5\Gammaffl clauses requires exponential size. Previous bounds applied only when the number of clauses was at most linear in the number of variables. For the pigeonhole principle our bound is a small improvement over previous bounds. Our proofs are more elementary than previous arguments, and establish a connection between Resolution proof size and maximum clause size.
Toward A Good Algorithm for Determining Unsatisfiability of Propositional Formulas
, 1995
"... We present progress toward an algorithm that provides short certificates of unsatisfiability with high probability when inputs are random instances of 3SAT. Such an algorithm would incorporate an approximation algorithm A for the 3Hitting Set problem. Using A it would determine an approximation ..."
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We present progress toward an algorithm that provides short certificates of unsatisfiability with high probability when inputs are random instances of 3SAT. Such an algorithm would incorporate an approximation algorithm A for the 3Hitting Set problem. Using A it would determine an approximation for the minimum fraction of variables that must be set to true (false) in order to satisfy the positive (negative) clauses. If the fraction is high enough, then the instance is deemed unsatisfiable. Key words Satisfiability, Resolution, Theorem Proving, Hitting Set 1 Introduction It is well known that the problem of determining the existence of a satisfying truth assignment for a given propositional formula in Conjunctive Normal Form (CNF) is NPcomplete. If clauses have exactly three literals each, the problem is called 3SAT and this problem is also NPcomplete. However, there exist polynomial time algorithms that, under certain circumstances, can produce a solution to a random satis...
Closure under replacements versus run time of DavisPutnam algorithms and distribution of satisfiable formulas
, 1996
"... We present an approach to characterize hard classes of formulas with respect to satisfiability by robustness under replacements. Besides some theoretical results we compare by means of experiments the distribution of satisfiable formulas in the constant clause length model, run time of a DavisPutna ..."
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We present an approach to characterize hard classes of formulas with respect to satisfiability by robustness under replacements. Besides some theoretical results we compare by means of experiments the distribution of satisfiable formulas in the constant clause length model, run time of a DavisPutnam algorithm, and the distribution of formulas closed under a fixed number of replacements of literals by the complementary literals.
Chapter 1 On the Satisability and Maximum Satisability of Random 3CNF Formulas
"... We analyze the pure literal rule heuristic for computing a satisfying assignment to a random 3CNF formula with n variables. We show that the pure literal rule by itself nds satisfying assignments for almost all 3CNF formulas with up to 1:63n clauses, but it fails for more than 1:7n clauses. As an ..."
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We analyze the pure literal rule heuristic for computing a satisfying assignment to a random 3CNF formula with n variables. We show that the pure literal rule by itself nds satisfying assignments for almost all 3CNF formulas with up to 1:63n clauses, but it fails for more than 1:7n clauses. As an aside we show that the value of maximum satisability for random 3CNF formulas is tightly concentrated around its mean. 1
ÄEPtovoa tcj puslie isleaiai Algorithms for the Satisfiability (SAT) Problem: A Survey
"... ABSTRACT. 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 desi ..."
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ABSTRACT. 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 satisfiability problems. 1.