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64
Stochastic Constraint Programming
, 2000
"... To model decision problems involving uncertainty and probability, we propose stochastic constraint programming. Stochastic constraint programs contain both decision variables (which we can set) and stochastic variables (which follow some probability distribution), and combine together the best ..."
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Cited by 76 (7 self)
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To model decision problems involving uncertainty and probability, we propose stochastic constraint programming. Stochastic constraint programs contain both decision variables (which we can set) and stochastic variables (which follow some probability distribution), and combine together the best features of traditional constraint satisfaction, stochastic integer programming, and stochastic satisfiability. We give a semantics for stochastic constraint programs, and propose a number of complete algorithms and approximation procedures. Using these algorithms, we observe phase transition behavior in stochastic constraint programs. Interestingly, the cost of both optimization and satisfaction peaks in the satisfaction phase boundary. Finally, we discuss a number of extensions of stochastic constraint programming to relax various assumptions like the independence between stochastic variables. Introduction Many real world decision problems contain uncertainty. Data about event...
MAP Complexity Results and Approximation Methods
 IN PROCEEDINGS OF THE 18TH CONFERENCE ON UNCERTAINTY IN ARTIFICIAL INTELLIGENCE (UAI
, 2002
"... MAP is the problem of finding a most probable instantiation of a set of variables in a Bayesian network, given some evidence. MAP appears ..."
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Cited by 48 (2 self)
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MAP is the problem of finding a most probable instantiation of a set of variables in a Bayesian network, given some evidence. MAP appears
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 48 (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,
Towards efficient sampling: Exploiting random walk strategies
 In Proceedings of the AAAI Conference on Artificial Intelligence
, 2004
"... From a computational perspective, there is a close connection between various probabilistic reasoning tasks and the problem of counting or sampling satisfying assignments of a propositional theory. We consider the question of whether stateoftheart satisfiability procedures, based on random walk ..."
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Cited by 47 (6 self)
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From a computational perspective, there is a close connection between various probabilistic reasoning tasks and the problem of counting or sampling satisfying assignments of a propositional theory. We consider the question of whether stateoftheart satisfiability procedures, based on random walk strategies, can be used to sample uniformly or nearuniformly from the space of satisfying assignments. We first show that random walk SAT procedures often do reach the full set of solutions of complex logical theories. Moreover, by interleaving random walk steps with Metropolis transitions, we also show how the sampling becomes nearuniform.
Model counting: A new strategy for obtaining good bounds
 In 21st AAAI
, 2006
"... Model counting is the classical problem of computing the number of solutions of a given propositional formula. It vastly generalizes the NPcomplete problem of propositional satisfiability, and hence is both highly useful and extremely expensive to solve in practice. We present a new approach to mod ..."
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Cited by 45 (16 self)
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Model counting is the classical problem of computing the number of solutions of a given propositional formula. It vastly generalizes the NPcomplete problem of propositional satisfiability, and hence is both highly useful and extremely expensive to solve in practice. We present a new approach to model counting that is based on adding a carefully chosen number of socalled streamlining constraints to the input formula in order to cut down the size of its solution space in a controlled manner. Each of the additional constraints is a randomly chosen XOR or parity constraint on the problem variables, represented either directly or in the standard CNF form. Inspired by a related yet quite different theoretical study of the properties of XOR constraints, we provide a formal proof that with high probability, the number of XOR constraints added in order to bring the formula to the boundary of being unsatisfiable determines with high precision its model count. Experimentally, we demonstrate that this approach can be used to obtain good bounds on the model counts for formulas that are far beyond the reach of exact counting methods. In fact, we obtain the first nontrivial solution counts for very hard, highly structured combinatorial problem instances. Note that unlike other counting techniques, such as Markov Chain Monte Carlo methods, we are able to provide highconfidence guarantees on the quality of the counts obtained.
Complexity results and approximation strategies for map explanations
 Journal of Artificial Intelligence Research
, 2004
"... MAP is the problem of finding a most probable instantiation of a set of variables given evidence. MAP has always been perceived to be significantly harder than the related problems of computing the probability of a variable instantiation (Pr), or the problem of computing the most probable explanatio ..."
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Cited by 43 (3 self)
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MAP is the problem of finding a most probable instantiation of a set of variables given evidence. MAP has always been perceived to be significantly harder than the related problems of computing the probability of a variable instantiation (Pr), or the problem of computing the most probable explanation (MPE). This paper investigates the complexity of MAP in Bayesian networks. Specifically, we show that MAP is complete for NP PP and provide further negative complexity results for algorithms based on variable elimination. We also show that MAP remains hard even when MPE and Pr become easy. For example, we show that MAP is NPcomplete when the networks are restricted to polytrees, and even then can not be effectively approximated. Given the difficulty of computing MAP exactly, and the difficulty of approximating MAP while providing useful guarantees on the resulting approximation, we investigate best effort approximations. We introduce a generic MAP approximation framework. We provide two instantiations of the framework; one for networks which are amenable to exact inference (Pr), and one for networks for which even exact inference is too hard. This allows MAP approximation on networks that are too complex to even exactly solve the easier problems, Pr and MPE. Experimental results indicate that using these approximation algorithms provides much better solutions than standard techniques, and provide accurate MAP estimates in many cases. 1.
From sampling to model counting
 In Proc. IJCAI’07
, 2007
"... We introduce a new technique for counting models of Boolean satisfiability problems. Our approach incorporates information obtained from sampling the solution space. Unlike previous approaches, our method does not require uniform or nearuniform samples. It instead converts local search sampling wit ..."
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Cited by 39 (8 self)
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We introduce a new technique for counting models of Boolean satisfiability problems. Our approach incorporates information obtained from sampling the solution space. Unlike previous approaches, our method does not require uniform or nearuniform samples. It instead converts local search sampling without any guarantees into very good bounds on the model count with guarantees. We give a formal analysis and provide experimental results showing the effectiveness of our approach. 1
Constraint Solving in Uncertain and Dynamic Environments: A Survey
 Constraints
, 2005
"... Abstract. This article follows a tutorial, given by the authors on dynamic constraint solving at CP 2003 [87]. It aims at offering an overview of the main approaches and techniques that have been proposed in the domain of constraint satisfaction to deal with uncertain and dynamic environments. Keywo ..."
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Cited by 37 (4 self)
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Abstract. This article follows a tutorial, given by the authors on dynamic constraint solving at CP 2003 [87]. It aims at offering an overview of the main approaches and techniques that have been proposed in the domain of constraint satisfaction to deal with uncertain and dynamic environments. Keywords: constraint satisfaction problem, uncertainty, change, stability, robustness, flexibility
The inferential complexity of Bayesian and credal networks
 In Proceedings of the International Joint Conference on Artificial Intelligence
, 2005
"... This paper presents new results on the complexity of graphtheoretical models that represent probabilities (Bayesian networks) and that represent interval and set valued probabilities (credal networks). We define a new class of networks with bounded width, and introduce a new decision problem for Ba ..."
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Cited by 35 (9 self)
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This paper presents new results on the complexity of graphtheoretical models that represent probabilities (Bayesian networks) and that represent interval and set valued probabilities (credal networks). We define a new class of networks with bounded width, and introduce a new decision problem for Bayesian networks, the maximin a posteriori. We present new links between the Bayesian and credal networks, and present new results both for Bayesian networks (most probable explanation with observations, maximin a posteriori) and for credal networks (bounds on probabilities a posteriori, most probable explanation with and without observations, maximum a posteriori). 1
Stochastic Constraint Programming: A ScenarioBased Approach
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"... To model combinatorial decision problems involving uncertainty and probability, we introduce scenario based stochastic constraint programming. Stochastic constraint programs contain both decision variables, which we can set, and stochastic variables, which follow a discrete probability distribution. ..."
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Cited by 30 (4 self)
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To model combinatorial decision problems involving uncertainty and probability, we introduce scenario based stochastic constraint programming. Stochastic constraint programs contain both decision variables, which we can set, and stochastic variables, which follow a discrete probability distribution. We provide a semantics for stochastic constraint programs based on scenario trees. Using this semantics, we can compile stochastic constraint programs down into conventional (nonstochastic) constraint programs. This allows us to exploit the full power of existing constraint solvers. We have implemented this framework for decision making under uncertainty in stochastic OPL, a language which is based on the OPL constraint modelling language [Hentenryck et al., 1999]. To illustrate the potential of this framework, we model a wide range of problems in areas as diverse as portfolio diversification, agricultural planning and production/inventory management.