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16
Improvements To Propositional Satisfiability Search Algorithms
, 1995
"... ... quickly across a wide range of hard SAT problems than any other SAT tester in the literature on comparable platforms. On a Sun SPARCStation 10 running SunOS 4.1.3 U1, POSIT can solve hard random 400variable 3SAT problems in about 2 hours on the average. In general, it can solve hard nvariable ..."
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Cited by 168 (0 self)
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... quickly across a wide range of hard SAT problems than any other SAT tester in the literature on comparable platforms. On a Sun SPARCStation 10 running SunOS 4.1.3 U1, POSIT can solve hard random 400variable 3SAT problems in about 2 hours on the average. In general, it can solve hard nvariable random 3SAT problems with search trees of size O(2 n=18:7 ). In addition to justifying these claims, this dissertation describes the most significant achievements of other researchers in this area, and discusses all of the widely known general techniques for speeding up SAT search algorithms. It should be useful to anyone interested in NPcomplete problems or combinatorial optimization in general, and it should be particularly useful to researchers in either Artificial Intelligence or Operations Research.
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 131 (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...
A Fast Parallel SATSolver  Efficient Workload Balancing
, 1994
"... We present a fast parallel SATsolver on a message based MIMD machine. The input formula is dynamically divided into disjoint subformulas. Small subformulas are solved by a fast sequential SATsolver running on every processor, which is based on the DavisPutnam procedure with a special heuristic fo ..."
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Cited by 50 (3 self)
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We present a fast parallel SATsolver on a message based MIMD machine. The input formula is dynamically divided into disjoint subformulas. Small subformulas are solved by a fast sequential SATsolver running on every processor, which is based on the DavisPutnam procedure with a special heuristic for variable selection. The algorithm uses optimized data structures to modify boolean formulas. Additionally efficient workload balancing algorithms are used, to achieve a uniform distribution of workload among the processors. We consider the communication network topologies ddimensional processor grid and linear processor array. Tests with up to 256 processors have shown very good efficiencyvalues (> 0.95).
Average Case Results for Satisfiability Algorithms Under the Random Clause Width Model
 Annals of Mathematics and Artificial Intelligence
, 1995
"... In the probabilistic analysis of algorithms for the Satisfiability problem, the randomclausewidth model is one of the most popular for generating random instances. This model is parameterized and it is not difficult to show that virtually the entire parameter space is covered by a collection of ..."
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Cited by 9 (1 self)
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In the probabilistic analysis of algorithms for the Satisfiability problem, the randomclausewidth model is one of the most popular for generating random instances. This model is parameterized and it is not difficult to show that virtually the entire parameter space is covered by a collection of polynomial time algorithms that find solutions to random instances with probability tending to 1 as instance size increases. But finding a collection of polynomial average time algorithms that cover the parameter space has proved much harder and such results have spanned approximately ten years. However, it can now be said that virtually the entire parameter space is covered by polynomial average time algorithms. This paper relates dominant, exploitable properties of random formulas over the parameter space to mechanisms of polynomial average time algorithms. The probabilistic discussion of such properties is new; main averagecase results over the last ten years are reviewed. 1 Intr...
Backtracking and Probing
, 1993
"... : We analyze two algorithms for solving constraint satisfaction problems. One of these algorithms, Probe Order Backtracking, has an average running time much faster than any previously analyzed algorithm for problems where solutions are common. Probe Order Backtracking uses a probing assignment (a p ..."
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Cited by 6 (2 self)
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: We analyze two algorithms for solving constraint satisfaction problems. One of these algorithms, Probe Order Backtracking, has an average running time much faster than any previously analyzed algorithm for problems where solutions are common. Probe Order Backtracking uses a probing assignment (a preselected test assignment to unset variables) to help guide the search for a solution to a constraint satisfaction problem. If the problem is not satisfied when the unset variables are temporarily set to the probing assignment, the algorithm selects one of the relations that the probing assignment fails to satisfy and selects an unset variable from that relation. Then at each backtracking step it generates subproblems by setting the selected variable each possible way. It simplifies each subproblem, and tries the same technique on them. For random problems with v variables, t clauses, and probability p that a literal appears in a clause, the average time for Probe Order Backtracking is no m...
The Probability of Pure Literals
"... We describe an error in earlier probabilistic analyses of the pure literal heuristic as a procedure for solving kSAT . All probabilistic analyses are in the constant degree model in which a random instance C of kSAT consists of m clauses selected independently and uniformly (with replacement) from ..."
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We describe an error in earlier probabilistic analyses of the pure literal heuristic as a procedure for solving kSAT . All probabilistic analyses are in the constant degree model in which a random instance C of kSAT consists of m clauses selected independently and uniformly (with replacement) from the set of all kclauses over n variables. We provide a new analysis for k = 2. Specifically, we show with probability approaching 1 as m goes to 1 one can apply the pure literal rule repeatedly to a random instance of 2SAT until the number of clauses is "small" provided n=m ? 1. But if n=m ! 1, with probability approaching 1 if the pure literal rule is applied as much as possible, then at least m 1=5 clauses will remain. Keywords: 2SAT , constant degree model, DavisPutnam Procedure, pure literal (heuristic), probability of a pure literal 1 1
A History of Satisfiability
, 2009
"... 1.1. Preface: the concept of satisfiability Interest in Satisfiability is expanding for a variety of reasons, not in the least because nowadays more problems are being solved faster by SAT solvers than other means. This is probably because Satisfiability stands at the crossroads of logic, graph theo ..."
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Cited by 4 (0 self)
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1.1. Preface: the concept of satisfiability Interest in Satisfiability is expanding for a variety of reasons, not in the least because nowadays more problems are being solved faster by SAT solvers than other means. This is probably because Satisfiability stands at the crossroads of logic, graph theory, computer science, computer engineering, and operations research. Thus, many problems originating in one of these fields typically have multiple translations to Satisfiability and there exist many mathematical tools available to the SAT solver to assist in solving them with improved performance. Because of the strong links to so many fields, especially logic, the history of Satisfiability can best be understood as it unfolds with respect to its logic roots. Thus, in addition to timelining events specific to Satisfiability, the chapter follows the presence of Satisfiability in logic as it was developed to model human thought and scientific reasoning through its use in computer design and now as modeling tool for solving a variety of practical problems. In order to succeed in this, we must introduce many ideas that have arisen during numerous attempts to reason
Average Time for the Full Pure Literal Rule
"... The simplified pure literal algorithm solves satisfiability problems by choosing variables in a fixed order and then generating subproblems for various values of the chosen variable. If some value satisfies every relation that depends on the chosen variable, then only the subproblem for that preferr ..."
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Cited by 3 (2 self)
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The simplified pure literal algorithm solves satisfiability problems by choosing variables in a fixed order and then generating subproblems for various values of the chosen variable. If some value satisfies every relation that depends on the chosen variable, then only the subproblem for that preferred value is generated. Otherwise, a subproblem is generated for every value of the variable. The full pure literal algorithm chooses variables that have a preferred value before choosing those that do not. A recurrence equation is found for the average time used by the full pure literal rule algorithm when solving random conjunctive normal form satisfiability problems. The random problems are characterized by the number of variables (v), the number of clauses (t), and the probability that a literal is in a clause (p). An asymptotic lower bound analysis shows that running time is more than polynomial in v when t increases more rapidly than (ln v) 2 (when p is set to maximize the running ...
Convergence Properties of Optimization Algorithms for the Satisfiability (SAT) Problem
 IEEE Trans. on Computers
, 1996
"... : The satisfiability (SAT) problem is a basic problem in computing theory. Presently, an active area of research on SAT problem is to design efficient optimization algorithms for finding a solution for a satisfiable CNF formula. A new formulation, the Universal SAT problem model, which transforms t ..."
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Cited by 2 (1 self)
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: The satisfiability (SAT) problem is a basic problem in computing theory. Presently, an active area of research on SAT problem is to design efficient optimization algorithms for finding a solution for a satisfiable CNF formula. A new formulation, the Universal SAT problem model, which transforms the SAT problem on Boolean space into an optimization problem on real space has been developed [31, 35, 34, 32]. Many optimization techniques, such as the steepest descent method, Newton's method, and the coordinate descent method, can be used to solve the Universal SAT problem. In this paper, we prove that, when the initial solution is sufficiently close to the optimal solution, the steepest descent method has a linear convergence ratio fi ! 1, Newton's method has a convergence ratio of order two, and the convergence ratio of the steepest descent method is approximately (1 \Gamma fi=m) for the Universal SAT problem with m variables. An algorithm based on the coordinate descent method for the...