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27
A study of residual supports in arc consistency
 In Proceedings of IJCAI’07
, 2007
"... Abstract. In an Arc Consistency (AC) algorithm, a residual support, or residue, is a support that has been stored during a previous execution of the procedure which determines if a value is supported by a constraint. The point is that a residue is not guaranteed to represent a lower bound of the sma ..."
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Cited by 32 (17 self)
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Abstract. In an Arc Consistency (AC) algorithm, a residual support, or residue, is a support that has been stored during a previous execution of the procedure which determines if a value is supported by a constraint. The point is that a residue is not guaranteed to represent a lower bound of the smallest current support of a value. In this paper, we study the theoretical impact of exploiting residues with respect to the basic algorithm AC3. First, we prove that AC3r (AC3 with unidirectional residues) and AC3rm (AC3 with multidirectional residues) are optimal for low and high constraint tightness. Second, we show that when AC has to be maintained during a backtracking search (the wellknown MAC algorithm), MAC2001 presents, with respect to MAC3r and MAC3rm, an overhead in O(µed) per branch of the binary tree built by MAC, where µ denotes the number of refutations of the branch, e the number of constraints and d the greatest domain size of the constraint network. One consequence is that, MAC3r and MAC3rm admit a better worstcase time complexity than MAC2001 for a branch involving µ refutations when either µ> d 2 or µ> d and the tightness of any constraint is either low or high. Our experimental results clearly show that exploiting residues allows enhancing MAC and SAC algorithms. 1
Data structures for generalised arc consistency for extensional constraints
 In Proceedings of the Twenty Second Conference on Artificial Intelligence
, 2007
"... Extensional (table) constraints are an important tool for attacking combinatorial problems with constraint programming. Recently there has been renewed interest in fast propagation algorithms for these constraints. We describe the use of two alternative data structures for maintaining generalised ar ..."
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Cited by 32 (9 self)
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Extensional (table) constraints are an important tool for attacking combinatorial problems with constraint programming. Recently there has been renewed interest in fast propagation algorithms for these constraints. We describe the use of two alternative data structures for maintaining generalised arc consistency on extensional constraints. The first, the NextDifference list, is novel and has been developed with this application in mind. The second, the trie, is well known but its use in this context is novel. Empirical analyses demonstrate the efficiency of the resulting approaches, both in GACschema, and in the watchedliteral table constraint in Minion.
Optimization of simple tabular reduction for table constraints
 In Proceedings of CP’08
, 2008
"... Abstract. Table constraints play an important role within constraint programming. Recently, many schemes or algorithms have been proposed to propagate table constraints or/and to compress their representation. We show that simple tabular reduction (STR), a technique proposed by J. Ullmann to dynamic ..."
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Cited by 25 (12 self)
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Abstract. Table constraints play an important role within constraint programming. Recently, many schemes or algorithms have been proposed to propagate table constraints or/and to compress their representation. We show that simple tabular reduction (STR), a technique proposed by J. Ullmann to dynamically maintain the tables of supports, is very often the most efficient practical approach to enforce generalized arc consistency within MAC. We also describe an optimization of STR which allows limiting the number of operations related to validity checking or search of supports. Interestingly enough, this optimization makes STR potentially r times faster where r is the arity of the constraint(s). The results of an extensive experimentation that we have conducted with respect to random and structured instances indicate that the optimized algorithm we propose is usually around twice as fast as the original STR and can be up to one order of magnitude faster than previous stateoftheart algorithms on some series of instances. 1
Maintaining Generalized Arc Consistency on Ad Hoc rary Constraints
"... In many reallife problems, constraints are explicitly defined as a set of solutions. This ad hoc (table) representation uses exponential memory and makes support checking (for enforcing GAC) difficult. In this paper, we address both problems simultaneously by representing an ad hoc constraint with ..."
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Cited by 22 (1 self)
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In many reallife problems, constraints are explicitly defined as a set of solutions. This ad hoc (table) representation uses exponential memory and makes support checking (for enforcing GAC) difficult. In this paper, we address both problems simultaneously by representing an ad hoc constraint with a multivalued decision diagram (MDD), a memory efficient data structure that supports fast support search. We explain how to convert a table constraint into an MDD constraint and how to maintain GAC on the MDD constraint. Thanks to a sparse set data structure, our MDDbased GAC algorithm, mddc, achieves full incrementality in constant time. Our experiments on structured problems, car sequencing and stilllife, show that mddc is a fast GAC algorithm for ad hoc constraints. It can replace a Boolean sequence constraint [1], and scales up well for structural MDD constraints with 208 variables and 340984 nodes. We also show why it is possible for mddc to be faster than the stateoftheart generic GAC algorithms in [2–4]. Its efficiency on nonstructural ad hoc constraints is justified empirically.
Random constraint satisfaction: easy generation of hard (satisfiable) instances
 Artificial Intelligence
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An MDDbased generalized arc consistency algorithm for positive and negative table constraints and some global constraints, Constraints 15 (2
, 2010
"... Abstract. A table constraint is explicitly represented its set of solutions or nonsolutions. This ad hoc (or extensional) representation may require space exponential to the arity of the constraint, making enforcing GAC expensive. In this paper, we address the space and time inefficiencies simult ..."
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Cited by 19 (1 self)
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Abstract. A table constraint is explicitly represented its set of solutions or nonsolutions. This ad hoc (or extensional) representation may require space exponential to the arity of the constraint, making enforcing GAC expensive. In this paper, we address the space and time inefficiencies simultaneously by presenting the mddc constraint. mddc is a global constraint that represents its (non)solutions with a multivalued decision diagram (MDD). The MDDbased representation has the advantage that it can be exponentially smaller than a table. The associated GAC algorithm (called mddc) has time complexity linear to the size of the MDD, and achieves full incrementality in constant time. In addition, we show how to convert a positive or negative table constraint into an mddc constraint in time linear to the size of the table. Our experiments on structured problems, car sequencing and stilllife, show that mddc is also a fast GAC algorithm for some global constraints such as sequence and regular. We also show that mddc is faster than the stateoftheart generic GAC algorithms in [2–4] for table constraint. 1
Advisors for Incremental Propagation
 THIRTEENTH INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING
, 2007
"... While incremental propagation for global constraints is recognized to be important, little research has been devoted to how propagatorcentered constraint programming systems should support incremental propagation. This paper introduces advisors as a simple and efficient, yet widely applicable metho ..."
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Cited by 15 (2 self)
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While incremental propagation for global constraints is recognized to be important, little research has been devoted to how propagatorcentered constraint programming systems should support incremental propagation. This paper introduces advisors as a simple and efficient, yet widely applicable method for supporting incremental propagation in a propagatorcentered setting. The paper presents how advisors can be used for achieving different forms of incrementality and evaluates cost and benefit for several global constraints.
Enforcing Arc Consistency using Bitwise Operations
 CONSTRAINT PROGRAMMING LETTERS
, 2007
"... In this paper, we propose to exploit bitwise operations to speed up some important computations such as looking for a support of a value in a constraint, or determining if a value is substitutable by another one. Considering a computer equipped with a xbit CPU, one can then expect an increase of th ..."
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Cited by 9 (6 self)
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In this paper, we propose to exploit bitwise operations to speed up some important computations such as looking for a support of a value in a constraint, or determining if a value is substitutable by another one. Considering a computer equipped with a xbit CPU, one can then expect an increase of the performance by a coefficient up to x (which may be important, since x is equal to 32 or 64 in many current central units). To show the interest of enforcing arc consistency using bitwise operations, we introduce a new variant of AC3, denoted by AC3 bit, which can be used when constraints are (or can be) represented in extension. This new algorithm when embedded in MAC, is approximately two times more efficient than AC3 rm. Note that AC3 rm is a variant of AC3 which exploits the concept of residual supports and has been shown to be faster than AC2001.
Global constraints: A survey
 IN
, 2011
"... Constraint programming (CP) is mainly based on filtering algorithms; their association with global constraints is one of the main strengths of CP because they exploit the specific structure of each constraint. This chapter is an overview of these two techniques. A collection of the most frequently u ..."
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Cited by 7 (1 self)
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Constraint programming (CP) is mainly based on filtering algorithms; their association with global constraints is one of the main strengths of CP because they exploit the specific structure of each constraint. This chapter is an overview of these two techniques. A collection of the most frequently used global constraints is given and some filtering algorithms are detailed. In addition, we try to identify how filtering algorithms can be designed. At last, we identify some problems that deserve to be addressed in the future.
Abscon 112 Toward more Robustness
"... Abstract. This paper describes the three main improvements made to the solver Abscon 109 [9]. The new version, Abscon 112, is able to automatically break some variable symmetries, infer allDifferent constraints from cliques of variables that are pairwise irreflexive, and use an optimized version of ..."
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Cited by 6 (0 self)
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Abstract. This paper describes the three main improvements made to the solver Abscon 109 [9]. The new version, Abscon 112, is able to automatically break some variable symmetries, infer allDifferent constraints from cliques of variables that are pairwise irreflexive, and use an optimized version of the STR (Simple Tabular Reduction) technique initially introduced by J. Ullmann for table constraints. 1 From Local to Global Variable Symmetries In [10], we have proposed to automatically detect variable symmetries of CSP instances by computing for each constraint scope a partition exhibiting locally symmetrical variables. From this local information that can be obtained in polynomial time, we can build a socalled lsvgraph whose automorphisms correspond to (global) variable symmetries. Interestingly enough, our approach allows us to disregard the representation (extension, intension, global) of constraints. Besides, the size of the lsvgraph is linear wrt the number of constraints (and their arity). To break symmetries from the generators returned by a graph automorphism