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17
Reusing CSP propagators for QCSPs
 In Proceedings of CSCLP2006
, 2006
"... Abstract. Quantified Constraint Satisfaction Problems are considerably more difficult to solve than classical CSP and the pruning obtained by local consistency is of crucial importance. In this paper, instead of designing specific consistency operators for constraints w.r.t each possible quantific ..."
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Abstract. Quantified Constraint Satisfaction Problems are considerably more difficult to solve than classical CSP and the pruning obtained by local consistency is of crucial importance. In this paper, instead of designing specific consistency operators for constraints w.r.t each possible quantification pattern, we propose to build them by relying on classical existential propagators and a few analysis of some mathematical properties of the constraints. It allows to reuse a large set of constraints already carefully implemented in existing solvers. Moreover, multiple levels of consistency for quantified constraint can be defined by choosing which analysis to use. This can be used to control the complexity of the pruning effort. We also introduce QeCode, a fullfeatured publicly available quantified constraint solver, built on top of Gecode.
A solver for quantified Boolean and linear constraints
 In Proc. of Int. Symp. on Applied Computing (SAC). ACM
, 2007
"... We make a number of contributions to the understanding and practical resolution of quantified constraints. Unlike previous work in the CP literature that was essentially focused on constraints expressed as binary tables, we focus on Presburger Arithmetics, i.e., Boolean combinations of linear constr ..."
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Cited by 6 (2 self)
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We make a number of contributions to the understanding and practical resolution of quantified constraints. Unlike previous work in the CP literature that was essentially focused on constraints expressed as binary tables, we focus on Presburger Arithmetics, i.e., Boolean combinations of linear constraints. From a theoretical perspective, we clarify the problem of the treatment of universal quantifiers by proposing a “symmetric ” version of the notion of quantified consistency. This notion imposes to maintain two constraint stores, which will be used to reason on universal and existential variables, respectively. We then describe a branch & bound algorithm that integrates both forms of propagation. Its implementation is, to the best of our knowledge, the first CP solver for this class of quantified constraints. 1.
Algorithms for stochastic csps
 In Proceedings of the 12th International Conference on the Principles and Practice of Constraint Programming
, 2006
"... Abstract. The Stochastic CSP (SCSP) is a framework recently introduced by Walsh to capture combinatorial decision problems that involve uncertainty and probabilities. The SCSP extends the classical CSP by including both decision variables, that an agent can set, and stochastic variables that follow ..."
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Cited by 5 (0 self)
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Abstract. The Stochastic CSP (SCSP) is a framework recently introduced by Walsh to capture combinatorial decision problems that involve uncertainty and probabilities. The SCSP extends the classical CSP by including both decision variables, that an agent can set, and stochastic variables that follow a probability distribution and can model uncertain events beyond the agent’s control. So far, two approaches to solving SCSPs have been proposed; backtrackingbased procedures that extend standard methods from CSPs, and scenariobased methods that solve SCSPs by reducing them to a sequence of CSPs. In this paper we further investigate the former approach. We first identify and correct a flaw in the forward checking (FC) procedure proposed by Walsh. We also extend FC to better take advantage of probabilities and thus achieve stronger pruning. Then we define arc consistency for SCSPs and introduce an arc consistency algorithm that can handle constraints of any arity. 1
Solution directed backjumping for QCSP
 In Proc. of Int. Conf. on Principles and Practice of Constraint Programming (CP
, 2007
"... Abstract. In this paper we present new techniques for improving backtracking based Quantified Constraint Satisfaction Problem (QCSP) solvers. QCSP is a generalization of CSP in which variables are either universally or existentially quantified and these quantifiers can be alternated in arbitrary way ..."
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Abstract. In this paper we present new techniques for improving backtracking based Quantified Constraint Satisfaction Problem (QCSP) solvers. QCSP is a generalization of CSP in which variables are either universally or existentially quantified and these quantifiers can be alternated in arbitrary ways. Our main new technique is solution directed backjumping (SBJ). In analogue to conflict directed backjumping, SBJ allows the solver to backtrack out of solved subtrees without having to find all of the distinct solutions normally required to validate the universal variables. Experiments with the solver QCSPSolve demonstrate that SBJ can improve its performance on random instances by orders of magnitude. In addition to this contribution, we demonstrate that performing varying levels of propagation for universal vs. existential variables can also be useful for enhancing performance. Finally, we discuss some techniques that are technically interesting but do not yet yield empirical improvements. 1
Realtime Online Solving of Quantified CSPs
 In Proceedings of CP
, 2009
"... Abstract. We define Realtime Online solving of Quantified Constraint Satisfaction Problems (QCSPs) as a model for realtime online CSP solving. We use a combination of propagation, lookahead and heuristics and show how all three improve performance. For adversarial opponents we show that we can achie ..."
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Abstract. We define Realtime Online solving of Quantified Constraint Satisfaction Problems (QCSPs) as a model for realtime online CSP solving. We use a combination of propagation, lookahead and heuristics and show how all three improve performance. For adversarial opponents we show that we can achieve promising results through good lookahead and heuristics and that a version of alpha beta pruning performs strongly. For random opponents, we show that we can frequently achieve solutions even on problems which lack a winning strategy and that we can improve our success rate by using Existential Quantified Generalised Arc Consistency, a lower level of consistency than SQGAC, to maximise pruning without removing solutions. We also consider the power of the universal opponent and show that through good heuristic selection we can generate a significantly stronger player than a static heuristic provides. 1
Quantified Constraint Satisfaction Problems: from relaxations to explanations
 In Proc. of Int. Joint. Conf. on Artificial Intelligence (IJCAI
, 2007
"... The Quantified Constraint Satisfaction Problem (QCSP) is a generalisation of the classical CSP in which some of variables can be universally quantified. In this paper, we extend two wellknown concepts in classical constraint satisfaction to the quantified case: problem relaxation and explanation of ..."
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The Quantified Constraint Satisfaction Problem (QCSP) is a generalisation of the classical CSP in which some of variables can be universally quantified. In this paper, we extend two wellknown concepts in classical constraint satisfaction to the quantified case: problem relaxation and explanation of inconsistency. We show that the generality of the QCSP allows for a number of different forms of relaxation not available in classical CSP. Wefurther present an algorithm for computing a generalisation of conflictbased explanations of inconsistency for the QCSP. 1
Weighted constraint satisfaction problems with minmax quantifiers
 In: ICTAI’11
, 2011
"... Abstract—Soft constraints are functions returning costs, and are essential in modeling overconstrained and optimization problems. We are interested in tackling soft constrained problems with adversarial conditions. Aiming at generalizing the weighted and quantified constraint satisfaction framewor ..."
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Cited by 4 (3 self)
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Abstract—Soft constraints are functions returning costs, and are essential in modeling overconstrained and optimization problems. We are interested in tackling soft constrained problems with adversarial conditions. Aiming at generalizing the weighted and quantified constraint satisfaction frameworks, a Quantified Weighted Constraint Satisfaction Problem (QWCSP) consists of a set of finite domain variables, a set of soft constraints, and a min or max quantifier associated with each of these variables. We formally define QWCSP, and propose a complete solver which is based on alphabeta pruning. QWCSPs are useful special cases of QCOP/QCOP+, and can be solved as a QCOP/QCOP+. Restricting our attention to only QWCSPs, we show empirically that our proposed solving techniques can better exploit problem characteristics than those developed for QCOP/QCOP+. Experimental results confirm the feasibility and efficiency of our proposals.
Nonbinary Quantified CSP: Algorithms and Modelling
"... The Quantified Constraint Satisfaction Problem (QCSP) extends classical CSP in a way which allows reasoning about uncertainty. In this paper I present novel algorithms for solving QCSP. Firstly I present algorithms to perform constraint propagation on reified disjunction constraints of any length. T ..."
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Cited by 3 (0 self)
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The Quantified Constraint Satisfaction Problem (QCSP) extends classical CSP in a way which allows reasoning about uncertainty. In this paper I present novel algorithms for solving QCSP. Firstly I present algorithms to perform constraint propagation on reified disjunction constraints of any length. The algorithms make full use of quantifier information to provide a high level of consistency. Secondly I present a scheme to enforce the nonbinary pure value rule. This rule is capable of pruning universal variables. Following this, two problems are modelled in nonbinary QCSP: the game of Connect 4, and a variant of jobshop scheduling with uncertainty, in the form of machine faults. The job shop scheduling example incorporates probability bounding of scenarios (such that only fault scenarios above a probability threshold are considered) and optimization of the schedule makespan. These contribute to the art of modelling in QCSP, and are a proof of concept for applying QCSP methods to complex, realistic problems. Both models make use of the reified disjunction constraint, and the nonbinary pure value rule. The example problems are used to evaluate the QCSP algorithms presented in this paper, identifying strengths and weaknesses, and to compare them to other QCSP approaches.
Consistencies for ultraweak solutions in minimax weighted csps using the duality principle
, 2012
"... Abstract. Minimax Weighted Constraint Satisfaction Problems (formerly called Quantified Weighted CSPs) are a framework for modeling soft constrained problems with adversarial conditions. In this paper, we describe novel definitions and implementations of node, arc and full directional arc consisten ..."
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Abstract. Minimax Weighted Constraint Satisfaction Problems (formerly called Quantified Weighted CSPs) are a framework for modeling soft constrained problems with adversarial conditions. In this paper, we describe novel definitions and implementations of node, arc and full directional arc consistency notions to help reduce search space on top of the basic tree search with alphabeta pruning for solving ultraweak solutions. In particular, these consistencies approximate the lower and upper bounds of the cost of a problem by exploiting the semantics of the quantifiers and reusing techniques from both Weighted and Quantified CSPs. Lower bound computation employs standard estimation of costs in the subproblems used in alphabeta search. In estimating upper bounds, we propose two approaches based on the Duality Principle: duality of quantifiers and duality of constraints. The first duality amounts to changing quantifiers from min to max, while the second duality reuses the lower bound approximation functions on dual constraints to generate upper bounds. Experiments on three benchmarks comparing basic alphabeta pruning and the six consistencies from the two dualities are performed to confirm the feasibility and efficiency of our proposal.
T.W.K.: A value ordering heuristic for solving ultraweak solutions in minimax weighted csps
 In: ICTAI’12
, 2012
"... work for modeling soft constrained problems with adversarial conditions. In this paper, we study the effects of a value ordering heuristic in solving ultraweak solutions on top of the alphabeta tree search with constraint propagation. The value ordering heuristic is based on minimax heuristics fro ..."
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work for modeling soft constrained problems with adversarial conditions. In this paper, we study the effects of a value ordering heuristic in solving ultraweak solutions on top of the alphabeta tree search with constraint propagation. The value ordering heuristic is based on minimax heuristics from adversarial search, which selects values for variables according to the semantic of quantifiers by considering the problem as a twoplayer zerosum game. In practice, implementing the heuristic requires costs approximations, and we devise three heuristic variants: HUnary, HBinary, and HFullBinary to approximate costs. In particular, we observe that combining these heuristic variants with consistency notions can achieve a better efficiency and a further reduction of search space. We perform experiments on three benchmarks to compare the effects on applying these heuristic variants, and confirm the feasibility and efficiency of our proposal. Index Terms—constraint optimization, soft constraint satisfaction, value ordering heuristics, minimax game search I.