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51
Algorithms and Complexity Results for #SAT and Bayesian Inference
 IN 44TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS
, 2004
"... Bayesian inference is an important problem with numerous applications in probabilistic reasoning. Counting satisfying assignments is a closely related problem of fundamental theoretical importance. In this paper, we show that plain old DPLL equipped with memoization (an algorithm we call #DPLLCache) ..."
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Cited by 63 (7 self)
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Bayesian inference is an important problem with numerous applications in probabilistic reasoning. Counting satisfying assignments is a closely related problem of fundamental theoretical importance. In this paper, we show that plain old DPLL equipped with memoization (an algorithm we call #DPLLCache) can solve both of these problems with time complexity that is at least as good as stateoftheart exact algorithms, and that it can also achieve the best known timespace tradeoff. We then proceed to show that there are instances where #DPLLCache can achieve an exponential speedup over existing algorithms.
A Lightweight Component Caching Scheme for Satisfiability Solvers
 In 10th International Conference on Theory and Applications of Satisfiability Testing
, 2007
"... Abstract. We introduce in this paper a lightweight technique for reducing work repetition caused by non–chronological backtracking commonly practiced by DPLL–based SAT solvers. The presented technique can be viewed as a partial component caching scheme. Empirical evaluation of the technique reveals ..."
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Cited by 60 (1 self)
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Abstract. We introduce in this paper a lightweight technique for reducing work repetition caused by non–chronological backtracking commonly practiced by DPLL–based SAT solvers. The presented technique can be viewed as a partial component caching scheme. Empirical evaluation of the technique reveals significant improvements on a broad range of industrial instances. 1
A compiler for deterministic, decomposable negation normal form
 In AAAI02
"... We present a compiler for converting CNF formulas into deterministic, decomposable negation normal form (dDNNF). This is a logical form that has been identified recently and shown to support a number of operations in polynomial time, including clausal entailment; model counting, minimization and e ..."
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Cited by 54 (12 self)
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We present a compiler for converting CNF formulas into deterministic, decomposable negation normal form (dDNNF). This is a logical form that has been identified recently and shown to support a number of operations in polynomial time, including clausal entailment; model counting, minimization and enumeration; and probabilistic equivalence testing. dDNNFs are also known to be a superset of, and more succinct than, OBDDs. The polytime logical operations supported by dDNNFs are a subset of those supported by OBDDs, yet are sufficient for modelbased diagnosis and planning applications. We present experimental results on compiling a variety of CNF formulas, some generated randomly and others corresponding to digital circuits. A number of the formulas we were able to compile efficiently could not be similarly handled by some stateoftheart model counters, nor by some stateoftheart OBDD compilers.
Hybrid backtracking bounded by treedecomposition of constraint networks
 ARTIFICIAL INTELLIGENCE
, 2003
"... We propose a framework for solving CSPs based both on backtracking techniques and on the notion of treedecomposition of the constraint networks. This mixed approach permits us to define a new framework for the enumeration, which we expect that it will benefit from the advantages of two approaches: ..."
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Cited by 50 (14 self)
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We propose a framework for solving CSPs based both on backtracking techniques and on the notion of treedecomposition of the constraint networks. This mixed approach permits us to define a new framework for the enumeration, which we expect that it will benefit from the advantages of two approaches: a practical efficiency of enumerative algorithms and a warranty of a limited time complexity by an approximation of the treewidth of the constraint networks. Finally, experimental results allow us to show the advantages of this approach.
Value Elimination: Bayesian Inference via Backtracking Search
 IN UAI03
, 2003
"... We present Value Elimination, a new algorithm for Bayesian Inference. Given the same variable ordering information, Value Elimination can achieve performance that is within a constant factor of variable elimination or recursive conditioning, and on some problems it can perform exponentially bet ..."
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Cited by 49 (2 self)
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We present Value Elimination, a new algorithm for Bayesian Inference. Given the same variable ordering information, Value Elimination can achieve performance that is within a constant factor of variable elimination or recursive conditioning, and on some problems it can perform exponentially better, irrespective of the variable ordering used by these algorithms. Value Elimination
AND/OR branchandbound search for combinatorial optimization in graphical models
, 2008
"... We introduce a new generation of depthfirst BranchandBound algorithms that explore the AND/OR search tree using static and dynamic variable orderings for solving general constraint optimization problems. The virtue of the AND/OR representation of the search space is that its size may be far small ..."
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Cited by 38 (19 self)
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We introduce a new generation of depthfirst BranchandBound algorithms that explore the AND/OR search tree using static and dynamic variable orderings for solving general constraint optimization problems. The virtue of the AND/OR representation of the search space is that its size may be far smaller than that of a traditional OR representation, which can translate into significant time savings for search algorithms. The focus of this paper is on linear space search which explores the AND/OR search tree rather than the search graph and therefore make no attempt to cache information. We investigate the power of the minibucket heuristics within the AND/OR search space, in both static and dynamic setups. We focus on two most common optimization problems in graphical models: finding the Most Probable Explanation (MPE) in Bayesian networks and solving Weighted CSPs (WCSP). In extensive empirical evaluations we demonstrate that the new AND/OR BranchandBound approach improves considerably over the traditional OR search strategy and show how various variable ordering schemes impact the performance of the AND/OR search scheme.
On probabilistic inference by weighted model counting
 Artificial Intelligence
"... A recent and effective approach to probabilistic inference calls for reducing the problem to one of weighted model counting (WMC) on a propositional knowledge base. Specifically, the approach calls for encoding the probabilistic model, typically a Bayesian network, as a propositional knowledge base ..."
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Cited by 37 (1 self)
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A recent and effective approach to probabilistic inference calls for reducing the problem to one of weighted model counting (WMC) on a propositional knowledge base. Specifically, the approach calls for encoding the probabilistic model, typically a Bayesian network, as a propositional knowledge base in conjunctive normal form (CNF) with weights associated to each model according to the network parameters. Given this CNF, computing the probability of some evidence becomes a matter of summing the weights of all CNF models consistent with the evidence. A number of variations on this approach have appeared in the literature recently, that vary across three orthogonal dimensions. The first dimension concerns the specific encoding used to convert a Bayesian network into a CNF. The second dimensions relates to whether weighted model counting is performed using a search algorithm on the CNF, or by compiling the CNF into a structure that renders WMC a polytime operation in the size of the compiled structure. The third dimension deals with the specific properties of network parameters (local structure) which are captured in the CNF encoding. In this paper, we discuss recent work in this area across the above three dimensions, and demonstrate empirically its practical importance in significantly expanding the reach of exact probabilistic inference. We restrict our discussion to exact inference and model counting, even though other proposals have been extended for approximate inference and approximate model counting.
SampleSearch: Importance Sampling in Presence of Determinism
, 2009
"... The paper focuses on developing effective importance sampling algorithms for mixed probabilistic and deterministic graphical models. The use of importance sampling in such graphical models is problematic because it generates many useless zero weight samples which are rejected yielding an inefficient ..."
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Cited by 27 (4 self)
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The paper focuses on developing effective importance sampling algorithms for mixed probabilistic and deterministic graphical models. The use of importance sampling in such graphical models is problematic because it generates many useless zero weight samples which are rejected yielding an inefficient sampling process. To address this rejection problem, we propose the SampleSearch scheme that augments sampling with systematic constraintbased backtracking search. We characterize the bias introduced by the combination of search with sampling, and derive a weighting scheme which yields an unbiased estimate of the desired statistics (e.g. probability of evidence). When computing the weights exactly is too complex, we propose an approximation which has a weaker guarantee of asymptotic unbiasedness. We present results of an extensive empirical evaluation demonstrating that SampleSearch outperforms other schemes in presence of significant amount of determinism.
A structurebased variable ordering heuristic for SAT
 In Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence (IJCAI’03
, 2003
"... We propose a variable ordering heuristic for SAT, which is based on a structural analysis of the SAT problem. We show that when the heuristic is used by a DavisPutnam SAT solver that employs conflictdirected backtracking, it produces a divideandconquer behavior in which the SAT problem is recurs ..."
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Cited by 25 (2 self)
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We propose a variable ordering heuristic for SAT, which is based on a structural analysis of the SAT problem. We show that when the heuristic is used by a DavisPutnam SAT solver that employs conflictdirected backtracking, it produces a divideandconquer behavior in which the SAT problem is recursively decomposed into smaller problems that are solved independently. We discuss the implications of this divideandconquer behavior on our ability to provide structurebased guarantees on the complexity of DavisPutnam SAT solvers. We also report on the integration of this heuristic with ZChaff — a stateoftheart SAT solver—showing experimentally that it significantly improves performance on a range of benchmark problems that exhibit structure. 1
Approximate counting by sampling the backtrackfree search space
 In AAAI
, 2007
"... We present a new estimator for counting the number of solutions of a Boolean satisfiability problem as a part of an importance sampling framework. The estimator uses the recently introduced SampleSearch scheme that is designed to overcome the rejection problem associated with distributions having a ..."
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Cited by 24 (6 self)
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We present a new estimator for counting the number of solutions of a Boolean satisfiability problem as a part of an importance sampling framework. The estimator uses the recently introduced SampleSearch scheme that is designed to overcome the rejection problem associated with distributions having a substantial amount of determinism. We show here that the sampling distribution of SampleSearch can be characterized as the backtrackfree distribution and propose several schemes for its computation. This allows integrating SampleSearch into the importance sampling framework for approximating the number of solutions and also allows using SampleSearch for computing a lower bound measure on the number of solutions. Our empirical evaluation demonstrates the superiority of our new approximate counting schemes against recent competing approaches.