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23
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 160 (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 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...
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 113 (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.
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.
Solving MAXrSAT above a Tight Lower Bound
, 2010
"... We present an exact algorithm that decides, for every fixed r ≥ 2 in time O(m) + 2 O(k2) whether a given multiset of m clauses of size r admits a truth assignment that satisfies at least ((2 r − 1)m + k)/2 r clauses. Thus MaxrSat is fixedparameter tractable when parameterized by the number of sat ..."
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Cited by 18 (8 self)
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We present an exact algorithm that decides, for every fixed r ≥ 2 in time O(m) + 2 O(k2) whether a given multiset of m clauses of size r admits a truth assignment that satisfies at least ((2 r − 1)m + k)/2 r clauses. Thus MaxrSat is fixedparameter tractable when parameterized by the number of satisfied clauses above the tight lower bound (1 − 2 −r)m. This solves an open problem of Mahajan, Raman and Sikdar (J. Comput. System Sci., 75, 2009). Our algorithm is based on a polynomialtime data reduction procedure that reduces a problem instance to an equivalent algebraically represented problem with O(k 2) variables. This is done by representing the instance as an appropriate polynomial, and by applying a probabilistic argument combined with some simple tools from Harmonic analysis to show that if the polynomial cannot be reduced to one of size O(k 2), then there is a truth assignment satisfying the required number of clauses. We introduce a new notion of bikernelization from a parameterized problem to another one and apply it to prove that the abovementioned parameterized MaxrSat admits a polynomialsize kernel. Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of Max2Sat with m clauses has at least 3k variables after application of certain polynomial time reduction rules to it, then there is a truth assignment that satisfies at least (3m + k)/4 clauses. We also outline how the fixedparameter tractability and polynomialsize kernel results on MaxrSat can be extended to more general families of Boolean
Elimination Of Infrequent Variables Improves Average Case Performance Of Satisfiability Algorithms
 SIAM J. Comput
, 1991
"... . We consider preprocessing a random instance I of CNF Satisfiability in order to remove infrequent variables (those which appear once or twice in an instance) from I. The model used to generate random instances is the popular randomclausesize model with parametersn, the number of clauses, r, the ..."
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Cited by 16 (5 self)
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. We consider preprocessing a random instance I of CNF Satisfiability in order to remove infrequent variables (those which appear once or twice in an instance) from I. The model used to generate random instances is the popular randomclausesize model with parametersn, the number of clauses, r, the number of Boolean variables from which clauses are composed, and p, the probability that a variable appears in a clause as a positive (or negative) literal. It is shown that exhaustive search over such preprocessed instances runs in polynomial average time over a significantly larger parameter space than has been shown for any other algorithm under the randomclausesize model when n = r ffl , ffl ! 1, and pr ! p fflr ln(r). Specifically, the results are that random instances of Satisfiability are "easy" in the average case if n = r ffl , 2=3 ? ffl ? 0, and pr ! (ln(n)=4) 1=3 r 2=3\Gammaffl ; or n = r ffl , 1 ? ffl 2=3, pr ! (1 \Gamma ffl \Gamma ffi) ln(n)=ffl for any ffi ? 0...
Solving #SAT Using Vertex Covers
, 2006
"... We propose an exact algorithm for counting the models of propositional formulas in conjunctive normal form (CNF). Our algorithm is based on the detection of strong backdoor sets of bounded size; each instantiation of the variables of a strong backdoor set puts the given formula into a class of form ..."
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Cited by 11 (4 self)
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We propose an exact algorithm for counting the models of propositional formulas in conjunctive normal form (CNF). Our algorithm is based on the detection of strong backdoor sets of bounded size; each instantiation of the variables of a strong backdoor set puts the given formula into a class of formulas for which models can be counted in polynomial time. For the backdoor set detection we utilize an efficient vertex cover algorithm applied to a certain “obstruction graph ” that we associate with the given formula. This approach gives rise to a new hardness index for formulas, the clusteringwidth. Our algorithm runs in uniform polynomial time on formulas with bounded clusteringwidth. It is known that the number of models of formulas with bounded cliquewidth, bounded treewidth, or bounded branchwidth can be computed in polynomial time; these graph parameters are applied to formulas via certain (hyper)graphs associated with formulas. We show that clusteringwidth and the other parameters mentioned are incomparable: there are formulas with bounded clusteringwidth and arbitrarily large cliquewidth, treewidth, and branchwidth. Conversely, there are formulas with arbitrarily large clusteringwidth and bounded cliquewidth, treewidth, and branchwidth.
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...
DNABased Parallel Computation by "Counting"
, 1997
"... The potential of DNA as a truly parallel computing device is enormous. Solutionphase DNA chemistry, though not unlimited, provides the only currentlyavailable experimental system. Its practical feasibility, however, is controversial. We have sought to extend the feasibility and generality of DNA c ..."
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Cited by 8 (4 self)
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The potential of DNA as a truly parallel computing device is enormous. Solutionphase DNA chemistry, though not unlimited, provides the only currentlyavailable experimental system. Its practical feasibility, however, is controversial. We have sought to extend the feasibility and generality of DNA computing by a novel application of the theory of counting . The biochemically equivalent operation for DNA counting is well known. We propose a DNA algorithm that employs this new operation. We also present an implementation of this algorithm by a novel DNAchemical method. Preliminary computer simulations suggest that the algorithm can significantly reduce the DNA space complexity (i.e., the maximum number of DNA molecules that must be present in the test tube during computation) for solving 3SAT to O(2 0:4n ). If the observation is correct, our algorithm can solve 3SAT instances of size up to or exceeding 120 variables. 1 Introduction 1.1 Two major issues in DNA computing Adleman [Ad...
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...