International audienceWe propose a new definition for characterizing levels of consistency. A perspective is to provide new tools for classifying filtering algorithms, including incomplete algorithms based on the semantics of constraints

Petit, Thierry

HAL Mines Nantes

Intermediary Local ConsistenciesThierry PetitTo cite this version:Thierry Petit. Intermediary Local Consistencies. European Conference on Artificial Intelligence(ECAI’12), Aug 2012, Montpellier, France, France. 2012. <hal-00753664>HAL Id: hal-00753664https://hal.inria.fr/hal-00753664Submitted on 19 Nov 2012HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Intermediary Local ConsistenciesThierry Petit1Abstract. We propose a new definition for characterizing levelsof consistency. A perspective is to provide new tools for classifyingfiltering algorithms, including incomplete algorithms based on thesemantics of constraints.1 INTRODUCTIONA filtering algorithm associated with a constraint is complete if itachieves generalized arc-consistency (GAC). This means that the al-gorithm removes from domains of variables all the values which can-not belong to at least one valid assignment satisfying the constraint.Depending on the size of problems and the nature of constraints, itis not always possible to enforce GAC. For instance, if a constraintinvolves n = 500 variables, using a O(n3) GAC algorithm is gen-erally not reasonable. The same remark can be made for algorithmsusing too heavy data structures. In this context, constraints are as-sociated with algorithms that enforce weaker forms of consistency.These intermediary consistencies are not always clearly formalized.This article provides a new generic definition for characterizingthe levels of consistency associated with constraints. We discusssome practical examples and identify the metrics that can be usedto classify the different levels of consistency.2 SUPPORT-DIRECTED CONSISTENCIESGiven a constraint C(X) defined on a set of variables X , a filter-ing algorithm removes values which, given the current variable do-mains, cannot belong to a solution satisfying the constraint. Somefiltering algorithms evaluate all the values in domains (e.g., GAC),while some other evaluate only the bounds of each domain, e.g.,Bounds-Consistency (BC). In both cases, the viability of a value vin the domain D(x) of a variable x ∈ X is checked by consideringeither the set of solutions of C(X) according to the current domains,or a superset of these solutions. We call this superset a relaxationof C(X). This relaxation can be obtained either by relaxing the con-straintC(X) itself, or by adding “virtually” some values in domains,for instance by considering that domains have no holes. If value vcannot belong to at least one solution of the relaxation then it canbe removed from D(x). Such a solution is called a support. Thus,two main notions characterize the level of consistency of a filteringalgorithm: 1. The set of checked values (either all the values in do-mains or only the bounds). 2. The relaxation of the constraint usedto search supports for values. We represent this relaxation by a setof constraints. This point of view leads to a new definition of localconsistency, parameterized by the relaxation. We use the notationsmin(x) = min(D(x)) and max(x) = max(D(x)).1 TASC (Mines Nantes, LINA, CNRS, INRIA), 4, Rue Alfred Kastler, FR-44307 Nantes Cedex 3, France, email: Thierry.Petit@mines-nantes.frDefinition 1 Given a constraintC(X), let Y = {y1, y2, . . . , yn} bea set of variables one-to-one mapped with X = {x1, x2, . . . , xn},such that ∀xi ∈ X, ∀yi ∈ Y , D(xi) ⊆ D(yi). Let C ={C1(Y1), C2(Y2), . . . , Cm(Ym)} be a set of constraints such thatY1 ∪ Y2 . . . Ym ⊆ Y and C(Y )⇒ C1(Y1) ∧ C2(Y2) . . . Cm(Ym).Value v ∈ D(xi) has a (C, Y )-support on C(X) if and only if∀Cj(Yj) ∈ C, either the variable yi mapped with xi is not in Yjor yi ∈ Yj and Cj(Yj) has a solution with yi = v.Constraint C(X) is (C, Y )-DC ((C, Y )-Domain Consistent) if andonly if ∀xi ∈ X , ∀v ∈ D(xi), v has a (C, Y )-support on C(X).C(X) is (C, Y )-BC ((C, Y )-Bounds Consistent) if and only if ∀xi ∈X , min(xi) and max (xi) have a (C, Y )-support on C(X).Definition 1 can be specialized to the usual notions of GAC, BCand Range-Consistency (RC) [2].Property 1 C(X) is GAC ≡ C(X) is ({C(X)}, X)-DC. Let Y bea variable set one-to-one mapped with X , with ∀yi ∈ Y,D(yi) ={min(xi),min(xi) + 1, . . . ,max(xi)}. C(X) is RC ≡ C(X) is({C(Y )}, Y )-DC. C(X) is BC ≡ C(X) is ({C(Y )}, Y )-BC.Definition 1 is not restricted to the case where all the variables in Yare involved in constraints of C, but constraints in C use exclusivelyvariables derived from variables in X . Our goal is to characterizefiltering algorithms, not reformulations. We thus consider that a fil-tering algorithm does not add new variables to the problem. Othergeneric consistencies can be defined, for instance by relaxing onlya subset of variables in X , or by checking real supports (like GAC)only for the bounds (({C(X)}, X)-BC). Furthermore, Definition 1characterizes the level of consistency of some specialized filteringalgorithms, which are not always clearly formalized in the literature.Example 1 Consider the constraint s =∑xi∈X xi. GAC is NP-Hard.2 Conversely, enforcing GAC on∑xi∈X xi ≤ s is in P [14].Therefore, a possible consistency for s =∑xi∈X xi is (C, Y )-DCwith Y = X ∪{s} and C = {∑xi∈X xi ≤ s,∑xi∈X xi ≥ s}. ~In Example 1, the obtained level of consistency is equivalent to BC.This is not the case for some other filtering algorithms of constraintsthat use a similar principle of relaxation for checking supports, suchas s =∑xi∈X x2i , or the filtering algorithm of Cost-regular [3].With respect to soft global constraints [9, 6], the variable that mea-sures a violation degree is also generally filtered separately from itsminimum value and from its maximum value [5]. Many other exam-ples exist, some of them relax both the constraint and the variables.Comparison with Guido Tack’s Dissertation Tack [12] pro-posed a characterization of propagation levels. The notion of com-pleteness of domain approximations provides a classification. This2 Checking the satisfiability of k=∑xi∈X xi, where k is an integer, requiresin the general case to solve the NP-Hard Subset-sum problem [4, p. 223].characterization also considers the case of set variables, converselyto Definition 1. It is thus more generic. Definition 1 deals with a set ofconstraints C implied by the original constraint, which correspondsconcretely to many filtering algorithms of constraints, provided byexisting solvers. By evaluating properties of the set we obtain somenew and pragmatic measures for comparing local consistencies.3 PROPERTIES AND PERSPECTIVESWe present and discuss some metrics for classifying intermediarylocal consistencies, using Definition 1.A. Set of solutions The levels of consistency weaker than GACcannot be totally ordered with respect to the set of solutions of therelaxations of C(X) considered for checking supports. These sets ofare not necessarily included one another. However, some propertiesexist. Given to levels of consistency Φ1 and Φ2 applied to a constraintC(X), we say that Φ1 ≤ Φ2 when the set of values removed by Φ1from the domains of variables in X is included in the set of valuesremoved by Φ2. By construction, this relation is transitive.Property 2 If Y ⊆ Y ′ then (C, Y ′)-DC ≤ (C, Y )-DC and (C, Y ′)-BC ≤ (C, Y )-BC.Proof (Sketch): The same constraints in C are checked, with largerdomains in Y ′, compared with Y . Definition 1 imposes completechecks (that is, a value v that does not satisfies a constraintCj(Yj) ∈C cannot have a (C, Y )-support). The set of supports of each v ∈D(xi) with Y is included into the set obtained with Y ′. With respect to a comparison related to different sets C, recall thatin Definition 1 all Cj(Yj) ∈ C are considered separately. Therefore,the fact that the set of solutions of the constraint network defined by aset C is strictly included in the set of solutions obtained with anotherset of constraints is not sufficient, in the general case, to prove aninclusion. We thus consider a Berge-acyclic constraint network [1].Property 3 Given two sets of constraints C ={C1(Y1), C2(Y2), . . . , Cm(Ym)} and C′ ={C′1(Y ′1 ), C′2(Y ′2 ), . . . , C′m′(Y ′m′)}, such that Y1 ∪ Y2 . . . Ym =Y ′1 ∪ Y ′2 . . . Y ′m′ ⊆ Y , If:C1(Y1) ∧ . . . ∧ Cm(Ym) is a Berge-acyclic constraint network,and C1(Y1) ∧ . . . ∧ Cm(Ym)⇒ C′1(Y ′1 ) ∧ . . . ∧ C′m′(Y ′m′),then (C′, Y )-DC ≤ (C, Y )-DC and (C′, Y )-BC ≤ (C, Y )-BC .Proof (Sketch): The set of solutions of the network NC = C1(Y1) ∧. . . ∧ Cm(Ym) defined by C is included in the one of C′. If NC isBerge-acyclic then any value v supported by all Cj(Yj)’s belongs toa solution ofNC . From Definition 1, the property holds. Example 2 The Alldiff(X) constraint is satisfied if and only if allthe variables in X are pairwise distinct. In this example, the set Yis defined as in Property 1, n is the size of X and d is the max-imum domain size. In the literature, we find some filtering algo-rithms for Alldiff(X): GAC (({C(X)}, X)-DC) in O(n1.5d) [11].RC (({C(Y )}, Y )-DC) in O(n2) [7]. BC (({C(Y )}, Y )-BC) inO(n. log(n)) [10, 8]. Some constraint toolkits such as Choco havea propagator that simulates a clique of binary constraints of differ-ence: (C6=, X)-DC with C6= = {xi 6= xj , i ∈ {1, 2, . . . n}, j ∈{1, 2, . . . n}, i 6= j}). Its has an O(n2) time complexity per branchof the search tree (it only reacts on variable assignments), whichleads to an average time complexity per node in O(n). We have:({C(Y )}, Y )-BC ≤ ({C(Y )}, Y )-DC ≤ ({C(X)}, X)-DC.(C6=, X)-DC ≤ ({C(X)}, X)-DC.Observe that since C6= does not corresponds to a Berge-acyclic con-straint network, the second inclusion cannot be reversed. ~It is possible to compose several propagators for C(X). Given twolevels of consistency Φ1 and Φ2, Φ1 ≤ Φ1 ◦Φ2 and Φ2 ≤ Φ1 ◦Φ2.B. Time complexity Given two levels of consistency Φ1 and Φ2applied to a constraint C(X), we say that Φ1 > Φ2 when thebest known algorithm for achieving Φ1 has a time complexity stricltygreater than the best known algorithm for achieving Φ2. This notionis not formal but very useful in practice to deal with large problems.Example 3 Consider the Alldiff(X) constraint of Example 2. Likein the example 2, the set Y is defined as in Property 1. We have:({C(X)}, X)-DC > ({C(Y )}, Y )-BC({C(Y )}, Y )-DC > ({C(Y )}, Y )-BCand ({C(Y )}, Y )-BC > (C6=, X)-DC. ~Time complexity is however not always the single criterion in thecontext of large problems. Some algorithms involve too complex datastructures. In some cases, domains of variables must be representedonly by their bounds, for memory reasons (trail). Thus, even when aGAC algorithm has a reasonable time complexity, e.g., less than orequal to O(n. log(n)), implementing a BC algorithm can be useful.C. Solution Counting The sets of solutions of distinct relaxationsof C(X) are not necessarily comparable. Conversely, the number ofsolutions of such relaxations can be ordered. Techniques for evaluat-ing the solution counting information of constraints [13] could be aninteresting metric to classify, possibly dynamically, several levels ofconsistencies characterized with Definition 1 for a given constraint.REFERENCES[1] C. Berge, ‘Graphs and hypergraphs’, Dunod, Paris, (1970).[2] C. Bessie`re, ‘Constraint propagation’, Research report 06020 (Chapter3 of the Handbook of Constraint Programming), LIRMM, (2006).[3] S. Demassey, G. Pesant, and L.-M. Rousseau, ‘A cost-regular basedhybrid column generation approach’, Constraints, 11(4), 315–333,(2006).[4] M. R. Garey and D. S. Johnson, ‘Computers and intractability : A guideto the theory of NP-completeness’, W.H. Freeman and Company, ISBN0-7167-1045-5, (1979).[5] E. Hebrard, D. Marx, B. O’Sullivan, and I. Razgon, ‘Soft constraints ofdifference and equality’, J. Artif. Intell. Res. (JAIR), 41, 97–130, (2011).[6] W.-J. Van Hoeve, G. Pesant, and L.-M. Rousseau, ‘On global warming:Flow-based soft global constraints’, Journal of Heuristics, 12:4-5, 475–489, (2006).[7] M. Leconte, ‘A bounds-based reduction scheme for constraints of dif-ference’, Proc. Constraint-96 International Workshop on Constraint-Based Reasoning, Key West, Florida, 19–28, (1996).[8] A. Lo´pez-Ortiz, C.-G. Quimper, J. Tromp, and P. van Beek, ‘A fast andsimple algorithm for bounds consistency of the alldifferent constraint’,Proc. IJCAI, 245–250, (2003).[9] T. Petit, J-C. Re´gin, and C. Bessie`re, ‘Specific filtering algorithms forover constrained problems’, Proc. CP, 451–463, (2001).[10] J-F. Puget, ‘A fast algorithm for the bound consistency of alldiff con-straints.’, Proc. AAAI, 359–366, (1998).[11] J-C. Re´gin, ‘A filtering algorithm for constraints of difference in CSPs’,Proc. AAAI, 362–367, (1994).[12] Guido Tack, Constraint Propagation – Models, Techniques, Implemen-tation, Doctoral dissertation, Saarland University, 2009.[13] A. Zanarini and G. Pesant, ‘Solution counting algorithms for constraint-centered search heuristics’, Proc. CP, 743–757, (2007).[14] Y. Zhang and R. H. C. Yap, ‘Arc consistency on n-ary monotonic andlinear constraints’, Proc. CP, 470–483, (2000).

Intermediary Local Consistencies

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HAL Id: hal-00753664https://hal.inria.fr/hal-00753664Submitted on 19 Nov 2012HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Intermediary Local ConsistenciesThierry PetitTo cite this version:Thierry Petit. Intermediary Local Consistencies. European Conference on Artificial Intelligence(ECAI’12), Aug 2012, Montpellier, France, France. ￿hal-00753664￿Intermediary Local ConsistenciesThierry Petit1Abstract. We propose a new definition for characterizing levelsof consistency. A perspective is to provide new tools for classifyingfiltering algorithms, including incomplete algorithms based on thesemantics of constraints.1 INTRODUCTIONA filtering algorithm associated with a constraint is complete if itachieves generalized arc-consistency (GAC). This means that the al-gorithm removes from domains of variables all the values which can-not belong to at least one valid assignment satisfying the constraint.Depending on the size of problems and the nature of constraints, itis not always possible to enforce GAC. For instance, if a constraintinvolves n = 500 variables, using a O(n3) GAC algorithm is gen-erally not reasonable. The same remark can be made for algorithmsusing too heavy data structures. In this context, constraints are as-sociated with algorithms that enforce weaker forms of consistency.These intermediary consistencies are not always clearly formalized.This article provides a new generic definition for characterizingthe levels of consistency associated with constraints. We discusssome practical examples and identify the metrics that can be usedto classify the different levels of consistency.2 SUPPORT-DIRECTED CONSISTENCIESGiven a constraint C(X) defined on a set of variables X , a filter-ing algorithm removes values which, given the current variable do-mains, cannot belong to a solution satisfying the constraint. Somefiltering algorithms evaluate all the values in domains (e.g., GAC),while some other evaluate only the bounds of each domain, e.g.,Bounds-Consistency (BC). In both cases, the viability of a value vin the domain D(x) of a variable x ∈ X is checked by consideringeither the set of solutions of C(X) according to the current domains,or a superset of these solutions. We call this superset a relaxationof C(X). This relaxation can be obtained either by relaxing the con-straint C(X) itself, or by adding “virtually” some values in domains,for instance by considering that domains have no holes. If value vcannot belong to at least one solution of the relaxation then it canbe removed from D(x). Such a solution is called a support. Thus,two main notions characterize the level of consistency of a filteringalgorithm: 1. The set of checked values (either all the values in do-mains or only the bounds). 2. The relaxation of the constraint usedto search supports for values. We represent this relaxation by a setof constraints. This point of view leads to a new definition of localconsistency, parameterized by the relaxation. We use the notationsmin(x) = min(D(x)) and max(x) = max(D(x)).1 TASC (Mines Nantes, LINA, CNRS, INRIA), 4, Rue Alfred Kastler, FR-44307 Nantes Cedex 3, France, email: Thierry.Petit@mines-nantes.frDefinition 1 Given a constraint C(X), let Y = {y1, y2, . . . , yn} bea set of variables one-to-one mapped with X = {x1, x2, . . . , xn},such that ∀xi ∈ X, ∀yi ∈ Y , D(xi) ⊆ D(yi). Let C ={C1(Y1), C2(Y2), . . . , Cm(Ym)} be a set of constraints such thatY1 ∪ Y2 . . . Ym ⊆ Y and C(Y )⇒ C1(Y1) ∧ C2(Y2) . . . Cm(Ym).Value v ∈ D(xi) has a (C, Y )-support on C(X) if and only if∀Cj(Yj) ∈ C, either the variable yi mapped with xi is not in Yjor yi ∈ Yj and Cj(Yj) has a solution with yi = v.Constraint C(X) is (C, Y )-DC ((C, Y )-Domain Consistent) if andonly if ∀xi ∈ X , ∀v ∈ D(xi), v has a (C, Y )-support on C(X).C(X) is (C, Y )-BC ((C, Y )-Bounds Consistent) if and only if ∀xi ∈X , min(xi) and max (xi) have a (C, Y )-support on C(X).Definition 1 can be specialized to the usual notions of GAC, BCand Range-Consistency (RC) [2].Property 1 C(X) is GAC ≡ C(X) is ({C(X)}, X)-DC. Let Y bea variable set one-to-one mapped with X , with ∀yi ∈ Y,D(yi) ={min(xi),min(xi) + 1, . . . ,max(xi)}. C(X) is RC ≡ C(X) is({C(Y )}, Y )-DC. C(X) is BC ≡ C(X) is ({C(Y )}, Y )-BC.Definition 1 is not restricted to the case where all the variables in Yare involved in constraints of C, but constraints in C use exclusivelyvariables derived from variables in X . Our goal is to characterizefiltering algorithms, not reformulations. We thus consider that a fil-tering algorithm does not add new variables to the problem. Othergeneric consistencies can be defined, for instance by relaxing onlya subset of variables in X , or by checking real supports (like GAC)only for the bounds (({C(X)}, X)-BC). Furthermore, Definition 1characterizes the level of consistency of some specialized filteringalgorithms, which are not always clearly formalized in the literature.Example 1 Consider the constraint s =∑xi∈X xi. GAC is NP-Hard.2 Conversely, enforcing GAC on∑xi∈X xi ≤ s is in P [14].Therefore, a possible consistency for s =∑xi∈X xi is (C, Y )-DCwith Y = X ∪{s} and C = {∑xi∈X xi ≤ s,∑xi∈X xi ≥ s}. ~In Example 1, the obtained level of consistency is equivalent to BC.This is not the case for some other filtering algorithms of constraintsthat use a similar principle of relaxation for checking supports, suchas s =∑xi∈X x2i , or the filtering algorithm of Cost-regular [3].With respect to soft global constraints [9, 6], the variable that mea-sures a violation degree is also generally filtered separately from itsminimum value and from its maximum value [5]. Many other exam-ples exist, some of them relax both the constraint and the variables.Comparison with Guido Tack’s Dissertation Tack [12] pro-posed a characterization of propagation levels. The notion of com-pleteness of domain approximations provides a classification. This2 Checking the satisfiability of k=∑xi∈X xi, where k is an integer, requiresin the general case to solve the NP-Hard Subset-sum problem [4, p. 223].characterization also considers the case of set variables, converselyto Definition 1. It is thus more generic. Definition 1 deals with a set ofconstraints C implied by the original constraint, which correspondsconcretely to many filtering algorithms of constraints, provided byexisting solvers. By evaluating properties of the set we obtain somenew and pragmatic measures for comparing local consistencies.3 PROPERTIES AND PERSPECTIVESWe present and discuss some metrics for classifying intermediarylocal consistencies, using Definition 1.A. Set of solutions The levels of consistency weaker than GACcannot be totally ordered with respect to the set of solutions of therelaxations of C(X) considered for checking supports. These sets ofare not necessarily included one another. However, some propertiesexist. Given to levels of consistency Φ1 and Φ2 applied to a constraintC(X), we say that Φ1 ≤ Φ2 when the set of values removed by Φ1from the domains of variables in X is included in the set of valuesremoved by Φ2. By construction, this relation is transitive.Property 2 If Y ⊆ Y ′ then (C, Y ′)-DC ≤ (C, Y )-DC and (C, Y ′)-BC ≤ (C, Y )-BC.Proof (Sketch): The same constraints in C are checked, with largerdomains in Y ′, compared with Y . Definition 1 imposes completechecks (that is, a value v that does not satisfies a constraint Cj(Yj) ∈C cannot have a (C, Y )-support). The set of supports of each v ∈D(xi) with Y is included into the set obtained with Y ′. With respect to a comparison related to different sets C, recall thatin Definition 1 all Cj(Yj) ∈ C are considered separately. Therefore,the fact that the set of solutions of the constraint network defined by aset C is strictly included in the set of solutions obtained with anotherset of constraints is not sufficient, in the general case, to prove aninclusion. We thus consider a Berge-acyclic constraint network [1].Property 3 Given two sets of constraints C ={C1(Y1), C2(Y2), . . . , Cm(Ym)} and C′ ={C′1(Y ′1 ), C′2(Y ′2 ), . . . , C′m′(Y ′m′)}, such that Y1 ∪ Y2 . . . Ym =Y ′1 ∪ Y ′2 . . . Y ′m′ ⊆ Y , If:C1(Y1) ∧ . . . ∧ Cm(Ym) is a Berge-acyclic constraint network,and C1(Y1) ∧ . . . ∧ Cm(Ym)⇒ C′1(Y ′1 ) ∧ . . . ∧ C′m′(Y ′m′),then (C′, Y )-DC ≤ (C, Y )-DC and (C′, Y )-BC ≤ (C, Y )-BC .Proof (Sketch): The set of solutions of the network NC = C1(Y1) ∧. . . ∧ Cm(Ym) defined by C is included in the one of C′. If NC isBerge-acyclic then any value v supported by all Cj(Yj)’s belongs toa solution ofNC . From Definition 1, the property holds. Example 2 The Alldiff(X) constraint is satisfied if and only if allthe variables in X are pairwise distinct. In this example, the set Yis defined as in Property 1, n is the size of X and d is the max-imum domain size. In the literature, we find some filtering algo-rithms for Alldiff(X): GAC (({C(X)}, X)-DC) in O(n1.5d) [11].RC (({C(Y )}, Y )-DC) in O(n2) [7]. BC (({C(Y )}, Y )-BC) inO(n. log(n)) [10, 8]. Some constraint toolkits such as Choco havea propagator that simulates a clique of binary constraints of differ-ence: (C6=, X)-DC with C6= = {xi 6= xj , i ∈ {1, 2, . . . n}, j ∈{1, 2, . . . n}, i 6= j}). Its has an O(n2) time complexity per branchof the search tree (it only reacts on variable assignments), whichleads to an average time complexity per node in O(n). We have:({C(Y )}, Y )-BC ≤ ({C(Y )}, Y )-DC ≤ ({C(X)}, X)-DC.(C6=, X)-DC ≤ ({C(X)}, X)-DC.Observe that since C6= does not corresponds to a Berge-acyclic con-straint network, the second inclusion cannot be reversed. ~It is possible to compose several propagators for C(X). Given twolevels of consistency Φ1 and Φ2, Φ1 ≤ Φ1 ◦Φ2 and Φ2 ≤ Φ1 ◦Φ2.B. Time complexity Given two levels of consistency Φ1 and Φ2applied to a constraint C(X), we say that Φ1 > Φ2 when thebest known algorithm for achieving Φ1 has a time complexity stricltygreater than the best known algorithm for achieving Φ2. This notionis not formal but very useful in practice to deal with large problems.Example 3 Consider the Alldiff(X) constraint of Example 2. Likein the example 2, the set Y is defined as in Property 1. We have:({C(X)}, X)-DC > ({C(Y )}, Y )-BC({C(Y )}, Y )-DC > ({C(Y )}, Y )-BCand ({C(Y )}, Y )-BC > (C6=, X)-DC. ~Time complexity is however not always the single criterion in thecontext of large problems. Some algorithms involve too complex datastructures. In some cases, domains of variables must be representedonly by their bounds, for memory reasons (trail). Thus, even when aGAC algorithm has a reasonable time complexity, e.g., less than orequal to O(n. log(n)), implementing a BC algorithm can be useful.C. Solution Counting The sets of solutions of distinct relaxationsof C(X) are not necessarily comparable. Conversely, the number ofsolutions of such relaxations can be ordered. Techniques for evaluat-ing the solution counting information of constraints [13] could be aninteresting metric to classify, possibly dynamically, several levels ofconsistencies characterized with Definition 1 for a given constraint.REFERENCES[1] C. Berge, ‘Graphs and hypergraphs’, Dunod, Paris, (1970).[2] C. Bessière, ‘Constraint propagation’, Research report 06020 (Chapter3 of the Handbook of Constraint Programming), LIRMM, (2006).[3] S. Demassey, G. Pesant, and L.-M. Rousseau, ‘A cost-regular basedhybrid column generation approach’, Constraints, 11(4), 315–333,(2006).[4] M. R. Garey and D. S. Johnson, ‘Computers and intractability : A guideto the theory of NP-completeness’, W.H. Freeman and Company, ISBN0-7167-1045-5, (1979).[5] E. Hebrard, D. Marx, B. O’Sullivan, and I. Razgon, ‘Soft constraints ofdifference and equality’, J. Artif. Intell. Res. (JAIR), 41, 97–130, (2011).[6] W.-J. Van Hoeve, G. Pesant, and L.-M. Rousseau, ‘On global warming:Flow-based soft global constraints’, Journal of Heuristics, 12:4-5, 475–489, (2006).[7] M. Leconte, ‘A bounds-based reduction scheme for constraints of dif-ference’, Proc. Constraint-96 International Workshop on Constraint-Based Reasoning, Key West, Florida, 19–28, (1996).[8] A. López-Ortiz, C.-G. Quimper, J. Tromp, and P. van Beek, ‘A fast andsimple algorithm for bounds consistency of the alldifferent constraint’,Proc. IJCAI, 245–250, (2003).[9] T. Petit, J-C. Régin, and C. Bessière, ‘Specific filtering algorithms forover constrained problems’, Proc. CP, 451–463, (2001).[10] J-F. Puget, ‘A fast algorithm for the bound consistency of alldiff con-straints.’, Proc. AAAI, 359–366, (1998).[11] J-C. Régin, ‘A filtering algorithm for constraints of difference in CSPs’,Proc. AAAI, 362–367, (1994).[12] Guido Tack, Constraint Propagation – Models, Techniques, Implemen-tation, Doctoral dissertation, Saarland University, 2009.[13] A. Zanarini and G. Pesant, ‘Solution counting algorithms for constraint-centered search heuristics’, Proc. CP, 743–757, (2007).[14] Y. Zhang and R. H. C. Yap, ‘Arc consistency on n-ary monotonic andlinear constraints’, Proc. CP, 470–483, (2000).

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Intermediary Local Consistencies

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