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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 161 (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 127 (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...
Testing Heuristics: We Have It All Wrong
 Journal of Heuristics
, 1995
"... The competitive nature of most algorithmic experimentation is a source of problems that are all too familiar to the research community. It is hard to make fair comparisons between algorithms and to assemble realistic test problems. Competitive testing tells us which algorithm is faster but not w ..."
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Cited by 119 (2 self)
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The competitive nature of most algorithmic experimentation is a source of problems that are all too familiar to the research community. It is hard to make fair comparisons between algorithms and to assemble realistic test problems. Competitive testing tells us which algorithm is faster but not why. Because it requires polished code, it consumes time and energy that could be spent doing more experiments. This paper argues that a more scientific approach of controlled experimentation, similar to that used in other empirical sciences, avoids or alleviates these problems. We have confused research and development; competitive testing is suited only for the latter. Most experimental studies of heuristic algorithms resemble track meets more than scientific endeavors. Typically an investigator has a bright idea for a new algorithm and wants to show that it works better, in some sense, than known algorithms. This requires computational tests, perhaps on a standard set of benchmark p...
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 114 (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.
G.: Logicbased benders decomposition
 Mathematical Programming
, 2003
"... Benders decomposition uses a strategy of “learning from one’s mistakes.” The aim of this paper is to extend this strategy to a much larger class of problems. The key is to generalize the linear programming dual used in the classical method to an “inference dual. ” Solution of the inference dual take ..."
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Cited by 45 (10 self)
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Benders decomposition uses a strategy of “learning from one’s mistakes.” The aim of this paper is to extend this strategy to a much larger class of problems. The key is to generalize the linear programming dual used in the classical method to an “inference dual. ” Solution of the inference dual takes the form of a logical deduction that yields Benders cuts. The dual is therefore very different from other generalized duals that have been proposed. The approach is illustrated by working out the details for propositional satisfiability and 01 programming problems. Computational tests are carried out for the latter, but the most promising contribution of logicbased Benders may be to provide a framework for combining optimization and constraint programming methods.
A Perspective on Certain Polynomial Time Solvable Classes of Satisfiability
 Discrete Applied Mathematics
, 1998
"... The scope of certain wellstudied polynomialtime solvable classes of Satisfiability is investigated relative to a polynomialtime solvable class consisting of what we call matched formulas. The class of matched formulas has not been studied in the literature, probably because it seems not to contai ..."
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Cited by 17 (2 self)
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The scope of certain wellstudied polynomialtime solvable classes of Satisfiability is investigated relative to a polynomialtime solvable class consisting of what we call matched formulas. The class of matched formulas has not been studied in the literature, probably because it seems not to contain many challenging formulas. Yet, we find that, in some sense, the matched formulas are more numerous than Horn, extended Horn, renamable Horn, qHorn, CCbalanced, or Single Lookahead Unit Resolution (SLUR) formulas. In addition, we find that relatively few unsatisfiable formulas are in any of the above classes. However, there are many unsatisfiable formulas that can be solved in polynomial time by restricting resolution so as not to generate resolvents of size greater than the number of literals in a maximum length clause. We use the wellstudied random kSAT model M(n;m;k) for generating CNF formulas with m clauses, each with k distinct literals, from n variables. We show, for all m=n ? 2...
Investigating a general hierarchy of polynomially decidable classes of CNF's based on short treelike resolution proofs
, 1999
"... We investigate a hierarchy Gk (U ; S) of classes of conjunctive normal forms, recognizable and SATdecidable in polynomial time, with special emphasize on the corresponding hardness parameter hU ;S (F ) for clausesets F (the first level of inclusion). At level 0 an (incomplete, polytime) oracl ..."
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Cited by 17 (10 self)
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We investigate a hierarchy Gk (U ; S) of classes of conjunctive normal forms, recognizable and SATdecidable in polynomial time, with special emphasize on the corresponding hardness parameter hU ;S (F ) for clausesets F (the first level of inclusion). At level 0 an (incomplete, polytime) oracle U for unsatisfiability detection and an oracle S for satisfiability detection is used. The hierarchy from [Pretolani 96] is improved in this way with respect to strengthened satisfiability handling, simplified recognition and consistent relativization. Also a hierarchy of canonical polytime reductions with Unitclause propagation at the first level is obtained. General methods for upper and lower bounds on hU ;S (F ) are developed and applied to a number of wellknown examples. hU ;S (F ) admits several different characterizations, including the space complexity of treelike resolution and the use of pebble games as in [Esteban, Tor'an 99]. Using for S the class of linearly sat...
On 2SAT and Renamable Horn
, 2000
"... We introduce new linear time algorithms for satisfiability of binary propositional theories (2SAT), and for recognition and satisfiability of renamable Horn theories. The algorithms are based on unit resolution, and are thus likely easier to integrate within general SAT solvers than other grap ..."
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Cited by 13 (3 self)
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We introduce new linear time algorithms for satisfiability of binary propositional theories (2SAT), and for recognition and satisfiability of renamable Horn theories. The algorithms are based on unit resolution, and are thus likely easier to integrate within general SAT solvers than other graphbased algorithms. Introduction 2SAT and renamable Horn SAT are the paradigmatic examples of tractable problems in propositional satisfiability, itself the paradigmatic example of NPcomplete problem. 2SAT is the problem of deciding the satisfiability of a set of binary clauses; renamable Horn SAT, or RenHornSAT for short, the problem of deciding the satisfiability of a set of clauses which is renamable Horn, i.e. which can be transformed into a set of Horn clauses by an uniform renaming of variables. We present linear time algorithms for satisfiability of binary theories, and for recognition and satisfiability of renamable Horn. For such two classic problems, the conceptual baggage, ...
Exact Selection of Minimal Unsatisfiable Subformulae for Special Classes of Propositional Formulae
 Annals of Mathematics and Artificial Intelligence
, 2002
"... A minimal unsatisfiable subformula (MUS) of a given propositional formula is a subset of clauses which is unsatisfiable, but becomes satisfiable as soon as we remove any of its clauses. In several applicative fields, a relevant problem besides solving the satisfiability problem, is to detect a minim ..."
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Cited by 10 (0 self)
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A minimal unsatisfiable subformula (MUS) of a given propositional formula is a subset of clauses which is unsatisfiable, but becomes satisfiable as soon as we remove any of its clauses. In several applicative fields, a relevant problem besides solving the satisfiability problem, is to detect a minimal unsatisfiable subformula. This is expecially true when the application is encoded into a propositional formula which should have a welldefined satisfiability property (either to be satisfiable or to be unsatisfiable). While selection of a MUS is a hard problem in general, we show classes of formulae for which such individuation can be performed e#ciently.
Detecting Embedded Horn Structure in Propositional Logic
 Information Processing Letters
, 1992
"... We show that the problem of finding a maximum renamable Horn problem within a propositional satisfiability problem is NPhard but can be formulated as a set packing and therefore a maximum clique problem, for which numerous algorithms and heuristics have been developed. 1 Introduction Horn clau ..."
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We show that the problem of finding a maximum renamable Horn problem within a propositional satisfiability problem is NPhard but can be formulated as a set packing and therefore a maximum clique problem, for which numerous algorithms and heuristics have been developed. 1 Introduction Horn clauses are widely used because, for them, satisfiability and inference problems are soluble in linear time. "Renamable Horn" problems (which are Horn up to a rescaling of variables) are also soluble in linear time. We address the problem of obtaining a renamable Horn problem by removing as few variables as possible from a given nonHorn satisfiability problem. One can then solve the original problem by enumerating truth assignments to the removed variables and solving a renamable Horn problem for each assignment. We show that finding a maximal renamable Horn subproblem can be formulated as a maximum clique problem, for which numerous algorithms and heuristics have been developed and tested [2,...