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30
An empirical investigation of value ordering for finding all solutions
 In Workshop on Modelling and Solving Problems with Constraints
, 2004
"... Abstract. Traditional backtracking search algorithms for solving constraint satisfaction problems select a variable and then construct a separate branch for each value in the variable’s domain: the order in which the values are assigned then has no effect on the overall search, when finding all solu ..."
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Abstract. Traditional backtracking search algorithms for solving constraint satisfaction problems select a variable and then construct a separate branch for each value in the variable’s domain: the order in which the values are assigned then has no effect on the overall search, when finding all solutions. Constraint solvers such as ILOG Solver and ECL i PS e instead use binary branching in constructing search trees: each node represents a choice between assigning a value to a variable and not assigning that value. The value ordering does then affect the search effort when finding all solutions. We show how this can happen by analysing search trees formed under opposite value orders and investigate empirically the effect on search from different value orders. The mechanism by which a good value order can prune the search suggests what we should aim for in choosing a value order. We suggest that for some classes of problem, lexicographic order may still be a priori a reasonable choice. 1
Efficient SAT techniques for absolute encoding of permutation problems: Application to Hamiltonian cycles
 IN: PROCEEDINGS SARA
, 2009
"... We study novel approaches for solving of hard combinatorial problems by translation to Boolean Satisfiability (SAT). Our focus is on combinatorial problems that can be represented as a permutation of n objects, subject to additional constraints. In the case of the Hamiltonian Cycle Problem (HCP), th ..."
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We study novel approaches for solving of hard combinatorial problems by translation to Boolean Satisfiability (SAT). Our focus is on combinatorial problems that can be represented as a permutation of n objects, subject to additional constraints. In the case of the Hamiltonian Cycle Problem (HCP), these constraints are that two adjacent nodes in a permutation should also be neighbors in the graph for which we search for a Hamiltonian cycle. We use the absolute SAT encoding of permutations, where for each of the n objects and each of its possible positions in a permutation, a predicate is defined to indicate whether the object is placed in that position. For implementation of this predicate, we compare the direct and logarithmic encodings that have been used previously, against 16 hierarchical parameterizable encodings of which we explore 416 instantiations. We propose the use of enumerative adjacency constraints—that enumerate the possible neighbors of a node in a permutation— instead of, or in addition to the exclusivity adjacency constraints—that exclude impossible neighbors, and that have been applied previously. We study 11 heuristics for efficiently choosing the first node in the Hamiltonian cycle, as well as 8 heuristics for static CNF variable ordering. We achieve at least 4 orders of magnitude average speedup on HCP benchmarks from the phase transition region, relative to the previously used encodings for solving of HCPs via SAT, such that the speedup is increasing with the size of the graphs.
Representations of Sets and Multisets in Constraint Programming
 In 4th International Workshop on Modelling and Reformulating Constraint Satisfaction Problems
, 2005
"... Abstract. Constraint programming is a powerful and general purpose tool, but its use is limited, as the process of refining a specification of a problem into an efficient constraint program (known as modelling) is more of an art than a science at present and must be learned by years of experience. T ..."
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Abstract. Constraint programming is a powerful and general purpose tool, but its use is limited, as the process of refining a specification of a problem into an efficient constraint program (known as modelling) is more of an art than a science at present and must be learned by years of experience. This paper theoretically analyses one frequently occurring pattern in modelling, how to choose between different representations of highlevel structures, in particular sets and multisets. It differs from previous work by providing methods of comparing very different representations, and by abstracting away from a particular implementation of a representation. It demonstrates useful theoretical dominance results between representations in both a problem dependant and independent context. 1 1
Search in the patience game ’black hole
, 2005
"... We propose card games for one player as a valuable domain for studying search problems. They are a natural AI problem, as they are a widely enjoyed recreation for which solving techniques are generally not studied. We focus on a particular patience, called Black Hole. We show that a general version ..."
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We propose card games for one player as a valuable domain for studying search problems. They are a natural AI problem, as they are a widely enjoyed recreation for which solving techniques are generally not studied. We focus on a particular patience, called Black Hole. We show that a general version of it is NPcomplete. Then we show that we can fruitfully study a number of mature AI paradigms applied to this single problem. An important feature of Black Hole is the presence of symmetries which arise during the search process, and we show that tacking these can improve search dramatically. Our empirical evaluation shows that Black Hole is winnable approximately 87 % of the time. 1
A constraint programming approach for solving a queueing design and control problem
 INFORMS Journal on Computing
, 2009
"... In a facility with front room and back room operations, it is useful to switch workers between the rooms in order to cope with changing customer demand. Assuming stochastic customer arrival and service times, we seek a policy for switching workers such that the expected customer waiting time is mini ..."
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In a facility with front room and back room operations, it is useful to switch workers between the rooms in order to cope with changing customer demand. Assuming stochastic customer arrival and service times, we seek a policy for switching workers such that the expected customer waiting time is minimized while the expected back room staffing is sufficient to perform all work. Three novel constraint programming models and several shaving procedures for these models are presented. Experimental results show that a model based on closedform expressions together with a combination of shaving procedures is the most efficient. This model is able to find and prove optimal solutions for many problem instances within a reasonable runtime. Previously, the only available approach was a heuristic algorithm. Furthermore, a hybrid method combining the heuristic and the best constraint programming method is shown to perform as well as the heuristic in terms of solution quality over time, while achieving the same performance in terms of proving optimality as the pure constraint programming model. This is the first work of which we are aware that solves such queueingbased problems with constraint programming. 1.
Solving a stochastic queueing control problem with constraint programming
, 2006
"... Abstract. In a facility with front room and back room operations, it is useful to switch workers between the rooms in order to cope with changing customer demand. Assuming stochastic customer arrival and service times, we seek a policy for switching workers such that the expected customer waiting t ..."
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Abstract. In a facility with front room and back room operations, it is useful to switch workers between the rooms in order to cope with changing customer demand. Assuming stochastic customer arrival and service times, we seek a policy for switching workers such that the expected customer waiting time is minimized while the expected back room staffing is sufficient to perform all work. Three novel constraint programming models and a shaving algorithm are presented. Experimental results show that the best constraint programming model, using shaving, is able to find and prove optimal solutions for almost all problem instances within a reasonable runtime, but that an existing heuristic algorithm performs better in terms of solution quality over time. A hybrid method combining the heuristic and the best constraint programming method is shown to perform better than either of these approaches separately. This is the first work of which we are aware that solves a queueing control problem with constraint programming. 1
The Systematic Generation Of Channelled Models In Constraint Satisfaction
"... Key words: constraint modelling, channelling constraints, channels, channelled models, representations, redundant representations, automatic modelling, refinement Solving a problem with finitedomain constraint programming requires generating a model from the informal description of the problem such ..."
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Key words: constraint modelling, channelling constraints, channels, channelled models, representations, redundant representations, automatic modelling, refinement Solving a problem with finitedomain constraint programming requires generating a model from the informal description of the problem such that this model can be accepted by a constraint solver. This generation process, called constraint modelling, is considered a hard task due to the number of choices and decisions it includes. Experience of skilled modellers in handcrafting many effective models has allowed identifying numerous patterns. One of these patterns is the addition of redundant information to a model. When this addition takes place, the consistency between all the redundant information needs to be maintained. The special constraints inserted to carry out this consistency maintenance are called channelling constraints (channels) and the models
Speeding up weighted constraint satisfaction using redundant modeling
 IN: PROC. OF AI’06
, 2006
"... In classical constraint satisfaction, combining mutually redundant models using channeling constraints is effective in increasing constraint propagation and reducing search space for many problems. In this paper, we investigate how to benefit the same for weighted constraint satisfaction problems ( ..."
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In classical constraint satisfaction, combining mutually redundant models using channeling constraints is effective in increasing constraint propagation and reducing search space for many problems. In this paper, we investigate how to benefit the same for weighted constraint satisfaction problems (WCSPs), a common soft constraint framework for modeling optimization and overconstrained problems. First, we show how to generate a redundant WCSP model from an existing WCSP using generalized model induction. We then uncover why naively combining two WCSPs by posting channeling constraints as hard constraints and relying on the standard NC * and AC * propagation algorithms does not work well. Based on these observations, we propose mNC ∗ c and mAC ∗ c and their associated algorithms for effectively enforcing node and arc consistencies on a combined model with m submodels. The two notions are strictly stronger than NC * and AC * respectively. Experimental results confirm that applying the 2NC ∗ c and 2AC ∗ c algorithms on combined models reduces more search space and runtime than applying the stateoftheart AC*, FDAC*, and EDAC * algorithms on single models.
Automatic Generation of Redundant Models for Permutation Constraint Satisfaction Problems
"... If we have two representations of a problem as constraint satisfaction problem (CSP) models, it has been shown that combining the models using channeling constraints can increase constraint propagation in tree search CSP solvers. Handcrafting two CSP models for a problem, however, is often timecons ..."
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If we have two representations of a problem as constraint satisfaction problem (CSP) models, it has been shown that combining the models using channeling constraints can increase constraint propagation in tree search CSP solvers. Handcrafting two CSP models for a problem, however, is often timeconsuming. In this paper, we propose model induction, a process which generates a second CSP model from an existing model using channeling constraints, and study its theoretical properties. The generated induced model is in a different viewpoint, i.e., set of variables. It is mutually redundant to and can be combined with the input model, so that the combined model contains more redundant information, which is useful to increase constraint propagation. We also propose two methods of combining CSP models, namely model intersection and model channeling. The two methods allow combining two mutually redundant models in the same and different viewpoints respectively. We exploit the applications of model induction, intersection, and channeling and identify three new classes of combined models, which contain different amounts of redundant information. We construct combined models of Permutation CSPs and show in extensive benchmark results that the combined models are more robust and efficient to solve than the single models. 1
Scheduling with uncertain durations: generating βrobust schedules using constraint
"... programming ..."