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24
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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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
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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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
A PathOptimal GAC Algorithm for Table Constraints
 20TH EUROPEAN CONFERENCE ON ARTIFICIAL INTELLIGENCE (ECAI'12), FRANCE
, 2012
"... Filtering by Generalized Arc Consistency (GAC) is a fundamental technique in Constraint Programming. Recent advances in GAC algorithms for extensional constraints rely on direct manipulation of tables during search. Simple Tabular Reduction (STR), which systematically removes invalid tuples from ta ..."
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Cited by 5 (2 self)
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Filtering by Generalized Arc Consistency (GAC) is a fundamental technique in Constraint Programming. Recent advances in GAC algorithms for extensional constraints rely on direct manipulation of tables during search. Simple Tabular Reduction (STR), which systematically removes invalid tuples from tables, has been shown to be a simple yet efficient approach. STR2, a refinement of STR, is considered to be among the best filtering algorithms for positive table constraints. In this paper, we introduce a new GAC algorithm called STR3 that is specifically designed to enforce GAC during search. STR3 can completely avoid unnecessary traversal of tables, making it optimal along any path of the search tree. Our experiments show that STR3 is much faster than STR2 when the average size of the tables is not reduced drastically during search.
Efficient Algorithms for Singleton Arc Consistency
"... In this paper, we propose two original and efficient approaches for enforcing singleton arc consistency. In the first one, the data structures used to enforce arc consistency are shared between all subproblems where a domain is reduced to a singleton. This new algorithm is not optimal but it require ..."
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Cited by 4 (3 self)
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In this paper, we propose two original and efficient approaches for enforcing singleton arc consistency. In the first one, the data structures used to enforce arc consistency are shared between all subproblems where a domain is reduced to a singleton. This new algorithm is not optimal but it requires far less space and is often more efficient in practice than the optimal algorithm SACOpt. In the second approach, we perform several runs of a greedy search (where at each step, arc consistency is maintained), possibly detecting the singleton arc consistency of several values in one run. It is an original illustration of applying inference (i.e., establishing singleton arc consistency) by search. Using a greedy search allows benefiting from the incrementality of arc consistency, learning relevant information from conflicts and, potentially finding solution(s) during the inference process. We present extensive experiments that show the benefit of our two approaches.
SecondOrder Consistencies
"... In this paper, we propose a comprehensive study of secondorder consistencies (i.e., consistencies identifying inconsistent pairs of values) for constraint satisfaction. We build a full picture of the relationships existing between four basic secondorder consistencies, namely path consistency (PC), ..."
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Cited by 3 (2 self)
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In this paper, we propose a comprehensive study of secondorder consistencies (i.e., consistencies identifying inconsistent pairs of values) for constraint satisfaction. We build a full picture of the relationships existing between four basic secondorder consistencies, namely path consistency (PC), 3consistency (3C), dual consistency (DC) and 2singleton arc consistency (2SAC), as well as their conservative and strong variants. Interestingly, dual consistency is an original property that can be established by using the outcome of the enforcement of generalized arc consistency (GAC), which makes it rather easy to obtain since constraint solvers typically maintain GAC during search. On binary constraint networks, DC is equivalent to PC, but its restriction to existing constraints, called conservative dual consistency (CDC), is strictly stronger than traditional conservative consistencies derived from path consistency, namely partial path consistency (PPC) and conservative path consistency (CPC). After introducing a general algorithm to enforce strong (C)DC, we present the results of an experimentation over a wide range of benchmarks that demonstrate the interest of (conservative) dual consistency. In particular, we show that enforcing (C)DC before search clearly improves the performance of MAC (the algorithm that maintains GAC during search) on several binary and nonbinary structured problems. 1.
An Optimal Filtering Algorithm for Table Constraints
"... Abstract. Filtering algorithms for table constraints are constraintbased, which means that the propagation queue only contains information on the constraints that must be reconsidered. This paper proposes four efficient valuebased algorithms for table constraints, meaning that the propagation queu ..."
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Abstract. Filtering algorithms for table constraints are constraintbased, which means that the propagation queue only contains information on the constraints that must be reconsidered. This paper proposes four efficient valuebased algorithms for table constraints, meaning that the propagation queue also contains information on the removed values. One of these algorithms (AC5TCTr) is proved to have an optimal time complexity of O(r.t + r.d) per table constraint. Experimental results show that, on structured instances, all our algorithms are two or three times faster than the state of the art STR2+ and MDD c algorithms. 1
Encoding Table Constraints in CLP(FD) Based on Pairwise AC
"... Abstract. We present an implementation of table constraints in CLP(FD). For binary constraints, the supports of each value are represented as a finitedomain variable, and action rules are used to propagate value exclusions. The bitvector representation of finite domains facilitates constanttime r ..."
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Abstract. We present an implementation of table constraints in CLP(FD). For binary constraints, the supports of each value are represented as a finitedomain variable, and action rules are used to propagate value exclusions. The bitvector representation of finite domains facilitates constanttime removal of unsupported values. For nary constraints, we propose pairwise arc consistency (AC), which ensures that each value has a support in the domain of every related variable. Pairwise AC does not require introducing new problem variables as done in binarization methods and allows for compact representation of constraints. Nevertheless, pairwise AC is weaker than general arc consistency (GAC) in terms of pruning power and requires a final check when a constraint becomes ground. To remedy this weakness, we propose adopting early checks when constraints are sufficiently instantiated. Our experimentation shows that pairwise AC with early checking is as effective as GAC for positive constraints. 1
Short and Long Supports for Constraint Propagation
"... Specialpurpose constraint propagation algorithms frequently make implicit use of short supports — by examining a subset of the variables, they can infer support (a justification that a variablevalue pair may still form part of an assignment that satisfies the constraint) for all other variables an ..."
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Specialpurpose constraint propagation algorithms frequently make implicit use of short supports — by examining a subset of the variables, they can infer support (a justification that a variablevalue pair may still form part of an assignment that satisfies the constraint) for all other variables and values and save substantial work – but short supports have not been studied in their own right. The two main contributions of this paper are the identification of short supports as important for constraint propagation, and the introduction of HaggisGAC, an efficient and effective general purpose propagation algorithm for exploiting short supports. Given the complexity of HaggisGAC, we present it as an optimised version of a simpler algorithm ShortGAC. Although experiments demonstrate the efficiency of ShortGAC compared with other generalpurpose propagation algorithms where a compact set of short supports is available, we show theoretically and experimentally that HaggisGAC is even better. We also find that HaggisGAC performs better than GACSchema on fulllength supports. We also introduce a variant algorithm HaggisGACStable, which is adapted to avoid work on backtracking and in some cases can be faster and have significant reductions in memory use. All the proposed algorithms are excellent for propagating disjunctions of constraints. In all experiments with disjunctions we found our algorithms to be faster than Constructive Or and GACSchema by at least an order of magnitude, and up to three orders of magnitude. 1.
Global Inverse Consistency for Interactive Constraint Satisfaction ⋆
"... Abstract. Some applications require the interactive resolution of a constraint problem by a human user. In such cases, it is highly desirable that the person who interactively solves the problem is not given the choice to select values that do not lead to solutions. We call this property global inve ..."
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Abstract. Some applications require the interactive resolution of a constraint problem by a human user. In such cases, it is highly desirable that the person who interactively solves the problem is not given the choice to select values that do not lead to solutions. We call this property global inverse consistency. Existing systems simulate this either by maintaining arc consistency after each assignment performed by the user or by compiling offline the problem as a multivalued decision diagram. In this paper, we define several questions related to global inverse consistency and analyse their complexity. Despite their theoretical intractability, we propose several algorithms for enforcing global inverse consistency and we show that the best version is efficient enough to be used in an interactive setting on several configuration and design problems. We finally extend our contribution to the inverse consistency of tuples. 1
Bitvector Algorithms for Binary Constraint Satisfaction and Subgraph Isomorphism
"... A solution to a binary constraint satisfaction problem is a set of discrete values, one in each of a given set of domains, subject to constraints that allow only prescribed pairs of values in specified pairs of domains. Solutions are sought by backtrack search interleaved with a process that removes ..."
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A solution to a binary constraint satisfaction problem is a set of discrete values, one in each of a given set of domains, subject to constraints that allow only prescribed pairs of values in specified pairs of domains. Solutions are sought by backtrack search interleaved with a process that removes from domains those values that are currently inconsistent with provisional choices already made in the course of search. For each value in a given domain, a bitvector shows which values in another domain are or are not permitted in a solution. Bitvector representation of constraints allows bitparallel, therefore fast, operations for editing domains during search. This article revises and updates bitvector algorithms published in the 1970’s, and introduces focus search, which is a new bitvector algorithm relying more on search and less on domainediting than previous algorithms. Focus search is competitive within a limited family of constraint satisfaction problems. Determination of subgraph isomorphism is a specialized binary constraint satisfaction problem for which bitvector algorithms have been widely used since the 1980’s, particularly for matching molecular structures. This article very substantially updates the author’s 1976 subgraph isomorphism algorithm, and reports experimental results with random and reallife data.