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73
Understanding Random SAT: Beyond the ClausestoVariables Ratio
 In Proc. of CP04
"... It is well known that the ratio of the number of clauses to the number of variables in a random kSAT instance is highly correlated with the instance's empirical hardness. We consider the problem of identifying such features of random SAT instances automatically with machine learning. We describe ..."
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Cited by 44 (17 self)
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It is well known that the ratio of the number of clauses to the number of variables in a random kSAT instance is highly correlated with the instance's empirical hardness. We consider the problem of identifying such features of random SAT instances automatically with machine learning. We describe and analyze models for three SAT solverskcnfs, oksolver and satzand for two different distributions of instances: uniform random 3SAT with varying ratio of clausestovariables, and uniform random 3SAT with fixed ratio of clausestovariables.
Backbones and backdoors in satisfiability
 In Proceedings of the National Conference on Artificial Intelligence (AAAI
, 2005
"... We study the backbone and the backdoors of propositional satisfiability problems. We make a number of theoretical, algorithmic and experimental contributions. From a theoretical perspective, we prove that backbones are hard even to approximate. From an algorithmic perspective, we present a number of ..."
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Cited by 28 (1 self)
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We study the backbone and the backdoors of propositional satisfiability problems. We make a number of theoretical, algorithmic and experimental contributions. From a theoretical perspective, we prove that backbones are hard even to approximate. From an algorithmic perspective, we present a number of different procedures for computing backdoors. From an empirical perspective, we study the correlation between being in the backbone and in a backdoor. Experiments show that there tends to be very little overlap between backbones and backdoors. We also study problem hardness for the Davis Putnam procedure. Problem hardness appears to be correlated with the size of strong backdoors, and weakly correlated with the size of the backbone, but does not appear to be correlated to the size of weak backdoors nor their number. Finally, to isolate the effect of backdoors, we look at problems with no backbone.
Predicting Learnt Clauses Quality in Modern SAT Solvers
, 2009
"... Beside impressive progresses made by SAT solvers over the last ten years, only few works tried to understand why Conflict Directed Clause Learning algorithms (CDCL) are so strong and efficient on most industrial applications. We report in this work a key observation of CDCL solvers behavior on this ..."
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Cited by 27 (6 self)
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Beside impressive progresses made by SAT solvers over the last ten years, only few works tried to understand why Conflict Directed Clause Learning algorithms (CDCL) are so strong and efficient on most industrial applications. We report in this work a key observation of CDCL solvers behavior on this family of benchmarks and explain it by an unsuspected side effect of their particular Clause Learning scheme. This new paradigm allows us to solve an important, still open, question: How to designing a fast, static, accurate, and predictive measure of new learnt clauses pertinence. Our paper is followed by empirical evidences that show how our new learning scheme improves stateofthe art results by an order of magnitude on both SAT and UNSAT industrial problems.
Satisfiability Solvers
, 2008
"... The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and h ..."
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Cited by 24 (0 self)
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The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and hardware verification [29–31, 228], automatic test pattern generation [138, 221], planning [129, 197], scheduling [103], and even challenging problems from algebra [238]. Annual SAT competitions have led to the development of dozens of clever implementations of such solvers [e.g. 13,
Empirical Hardness Models: Methodology and a Case Study on Combinatorial Auctions
"... Is it possible to predict how long an algorithm will take to solve a previouslyunseen instance of an NPcomplete problem? If so, what uses can be found for models that make such predictions? This paper provides answers to these questions and evaluates the answers experimentally. We propose the use ..."
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Cited by 19 (6 self)
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Is it possible to predict how long an algorithm will take to solve a previouslyunseen instance of an NPcomplete problem? If so, what uses can be found for models that make such predictions? This paper provides answers to these questions and evaluates the answers experimentally. We propose the use of supervised machine learning to build models that predict an algorithm’s runtime given a problem instance. We discuss the construction of these models and describe techniques for interpreting them to gain understanding of the characteristics that cause instances to be hard or easy. We also present two applications of our models: building algorithm portfolios that outperform their constituent algorithms, and generating test distributions that emphasize hard problems. We demonstrate the effectiveness of our techniques in a case study of the combinatorial auction winner determination problem. Our experimental results show that we can build very accurate models of an algorithm’s running time, interpret our models, build an algorithm portfolio that strongly outperforms the best single algorithm, and tune a standard benchmark suite to generate much harder problem instances.
Detecting backdoor sets with respect to horn and binary clauses
 In SAT’04
, 2004
"... Abstract. We study the parameterized complexity of detecting backdoor sets for instances of the propositional satisfiability problem (SAT) with respect to the polynomially solvable classes horn and 2cnf. A backdoor set is a subset of variables; for a strong backdoor set, the simplified formulas res ..."
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Cited by 18 (4 self)
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Abstract. We study the parameterized complexity of detecting backdoor sets for instances of the propositional satisfiability problem (SAT) with respect to the polynomially solvable classes horn and 2cnf. A backdoor set is a subset of variables; for a strong backdoor set, the simplified formulas resulting from any setting of these variables is in a polynomially solvable class, and for a weak backdoor set, there exists one setting which puts the satisfiable simplified formula in the class. We show that with respect to both horn and 2cnf classes, the detection of a strong backdoor set is fixedparameter tractable (the existence of a set of size k for a formula of length N can be decided in time f(k)N O(1)), but that the detection of a weak backdoor set is W[2]hard, implying that this problem is not fixedparameter tractable. 1
Ten challenges redux: Recent progress in propositional reasoning and search
 In Proceedings of CP ’03
, 2003
"... Abstract. In 1997 we presented ten challenges for research on satisfiability testing [1]. In this paper we review recent progress towards each of these challenges, including our own work on the power of clause learning and randomized restart policies. 1 ..."
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Cited by 16 (1 self)
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Abstract. In 1997 we presented ten challenges for research on satisfiability testing [1]. In this paper we review recent progress towards each of these challenges, including our own work on the power of clause learning and randomized restart policies. 1
Tradeoffs in the complexity of backdoor detection
 In Principles and Practice of Constraint Programming  CP 2007
, 2007
"... Abstract. There has been considerable interest in the identification of structural properties of combinatorial problems that lead to efficient algorithms for solving them. Some of these properties are “easily ” identifiable, while others are of interest because they capture key aspects of stateoft ..."
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Cited by 15 (6 self)
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Abstract. There has been considerable interest in the identification of structural properties of combinatorial problems that lead to efficient algorithms for solving them. Some of these properties are “easily ” identifiable, while others are of interest because they capture key aspects of stateoftheart constraint solvers. In particular, it was recently shown that the problem of identifying a strong Horn or 2CNFbackdoor can be solved by exploiting equivalence with deletion backdoors, and is NPcomplete. We prove that strong backdoor identification becomes harder than NP (unless NP=coNP) as soon as the inconsequential sounding feature of empty clause detection (present in all modern SAT solvers) is added. More interestingly, in practice such a feature as well as polynomial time constraint propagation mechanisms often lead to much smaller backdoor sets. In fact, despite the worstcase complexity results for strong backdoor detection, we show that SatzRand is remarkably good at finding small strong backdoors on a range of experimental domains. Our results suggest that structural notions explored for designing efficient algorithms for combinatorial problems should capture both statically and dynamically identifiable properties. 1
The Backdoor Key: A Path to Understanding Problem Hardness
 IN PROC. OF THE 19TH NAT. CONF. ON AI
, 2004
"... We introduce our work on the backdoor key, a concept that shows promise for characterizing problem hardness in backtracking search algorithms. The general notion of backdoors was recently introduced to explain the source of heavytailed behaviors in backtracking algorithms (Williams, Gomes, & S ..."
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Cited by 15 (0 self)
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We introduce our work on the backdoor key, a concept that shows promise for characterizing problem hardness in backtracking search algorithms. The general notion of backdoors was recently introduced to explain the source of heavytailed behaviors in backtracking algorithms (Williams, Gomes, & Selman 2003a; 2003b). We describe empirical studies that show that the key faction, i.e., the ratio of the key size to the corresponding backdoor size, is a good predictor of problem hardness of ensembles and individual instances within an ensemble for structure domains with large key fraction.