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Classifying the complexity of constraints using finite algebras
 SIAM Journal on Computing
, 2005
"... Abstract. Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NPcomplete in general, but certain restrictions on the form of the constraints can ensure tractability. Here we show that any set of relations used to specify th ..."
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Cited by 181 (33 self)
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Abstract. Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NPcomplete in general, but certain restrictions on the form of the constraints can ensure tractability. Here we show that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and we explore how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra. Hence, we completely translate the problem of classifying the complexity of restricted constraint satisfaction problems into the language of universal algebra. We introduce a notion of “tractable algebra, ” and investigate how the tractability of an algebra relates to the tractability of the smaller algebras which may be derived from it, including its subalgebras and homomorphic images. This allows us to reduce significantly the types of algebras which need to be classified. Using our results we also show that if the decision problem associated with a given collection of constraint types can be solved efficiently, then so can the corresponding search problem. We then classify all finite strictly simple surjective algebras with respect to tractability, obtaining a dichotomy theorem which generalizes Schaefer’s dichotomy for the generalized satisfiability problem. Finally, we suggest a possible general algebraic criterion for distinguishing the tractable and intractable cases of the constraint satisfaction problem.
ConjunctiveQuery Containment and Constraint Satisfaction
 Journal of Computer and System Sciences
, 1998
"... Conjunctivequery containment is recognized as a fundamental problem in database query evaluation and optimization. At the same time, constraint satisfaction is recognized as a fundamental problem in artificial intelligence. What do conjunctivequery containment and constraint satisfaction have in c ..."
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Cited by 164 (14 self)
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Conjunctivequery containment is recognized as a fundamental problem in database query evaluation and optimization. At the same time, constraint satisfaction is recognized as a fundamental problem in artificial intelligence. What do conjunctivequery containment and constraint satisfaction have in common? Our main conceptual contribution in this paper is to point out that, despite their very different formulation, conjunctivequery containment and constraint satisfaction are essentially the same problem. The reason is that they can be recast as the following fundamental algebraic problem: given two finite relational structures A and B, is there a homomorphism h : A ! B? As formulated above, the homomorphism problem is uniform in the sense that both relational structures A and B are part of the input. By fixing the structure B, one obtains the following nonuniform problem: given a finite relational structure A, is there a homomorphism h : A ! B? In general, nonuniform tractability results do not uniformize. Thus, it is natural to ask: which tractable cases of nonuniform tractability results for constraint satisfaction and conjunctivequery containment do uniformize? Our main technical contribution in this paper is to show that several cases of tractable nonuniform constraint satisfaction problems do indeed uniformize. We exhibit three nonuniform tractability results that uniformize and, thus, give rise to polynomialtime solvable cases of constraint satisfaction and conjunctivequery containment.
Composing Schema Mappings: SecondOrder Dependencies to the Rescue
 In PODS
, 2004
"... A schema mapping is a specification that describes how data structured under one schema (the source schema) is to be transformed into data structured under a di#erent schema (the target schema). Schema mappings play a key role in numerous areas of database systems, including database design, informa ..."
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Cited by 159 (20 self)
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A schema mapping is a specification that describes how data structured under one schema (the source schema) is to be transformed into data structured under a di#erent schema (the target schema). Schema mappings play a key role in numerous areas of database systems, including database design, information integration, and model management. A fundamental problem in this context is composing schema mappings: given two successive schema mappings, derive a schema mapping between the source schema of the first and the target schema of the second that has the same e#ect as applying successively the two schema mappings.
Algebraic structures in combinatorial problems
 TECHNICAL REPORT, TECHNISCHE UNIVERSITAT DRESDEN
, 2001
"... ..."
Schema Mappings, Data Exchange, and Metadata Management
, 2005
"... Schema mappings are highlevel specifications that describe the relationship between database schemas. Schema mappings are prominent in several different areas of database management, including database design, information integration, data exchange, metadata management, and peertopeer data managem ..."
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Cited by 127 (11 self)
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Schema mappings are highlevel specifications that describe the relationship between database schemas. Schema mappings are prominent in several different areas of database management, including database design, information integration, data exchange, metadata management, and peertopeer data management systems. Our main aim in this paper is to present an overview of recent advances in data exchange and metadata management, where the schema mappings are between relational schemas. In addition, we highlight some research issues and directions for future work.
The Approximability of Constraint Satisfaction Problems
, 2000
"... ... oftheoptimizationtask. Here weconsiderfourpossiblegoals: MaxCSP(MinCSP)isthe classofproblemswherethegoalistondanassignment maximizingthenumberofsatised factionproblemsdependingonthenatureofthe "underlying" constraintsaswellasonthegoal constraints(minimizingthenumberofunsatisedconstrain ..."
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Cited by 84 (1 self)
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... oftheoptimizationtask. Here weconsiderfourpossiblegoals: MaxCSP(MinCSP)isthe classofproblemswherethegoalistondanassignment maximizingthenumberofsatised factionproblemsdependingonthenatureofthe "underlying" constraintsaswellasonthegoal constraints(minimizingthenumberofunsatisedconstraints). MaxOnes(MinOnes)isthe classofoptimizationproblemswherethegoalistondan assignmentsatisfyingallconstraints withmaximum(minimum)numberofvariablesset to 1. Eachclassconsistsofinnitelymany thatdescribethepossibleconstraintsthatmaybeused. problemsandaproblemwithinaclass is specified by a finite collectionofniteBooleanfunctions pletelyclassiesalloptimizationproblems derived from Booleanconstraintsatisfaction.Our Creignou [11]. Inthisworkwedeterminetightboundsonthe "approximability"(i.e.,thera in MaxOnes,MinCSPandMinOnes.Combinedwiththeresultof Creignou,thiscomtiotowithinwhicheachproblemmay be approximatedinpolynomialtime)ofeveryproblem Tightboundsontheapproximabilityofeveryproblemin MaxCSPwereobtainedby resultscaptureadiversecollectionofoptimization problemssuchasMAX3SAT,MaxCut, (in)approximabilityoftheseoptimizationproblems andyieldacompactpresentationofmost MaxClique,MinCut,NearestCodewordetc. Ourresultsunifyrecentresultsonthe knownresults. Moreover, theseresultsprovideaformalbasistomanystatementsonthe behaviorofnaturaloptimizationproblems,thathaveso faronlybeenobservedempirically.
On the fixed parameter complexity of graph enumeration problems definable in monadic secondorder logic
, 2001
"... ..."
Constraint Satisfaction, Bounded Treewidth, and FiniteVariable Logics
, 2002
"... We systematically investigate the connections between constraint satisfaction problems, structures of bounded treewidth, and definability in logics with a finite number of variables. We first show that constraint satisfaction problems on inputs of treewidth less than k are definable using Datalog ..."
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Cited by 71 (12 self)
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We systematically investigate the connections between constraint satisfaction problems, structures of bounded treewidth, and definability in logics with a finite number of variables. We first show that constraint satisfaction problems on inputs of treewidth less than k are definable using Datalog programs with at most k variables; this provides a new explanation for the tractability of these classes of problems. After this, we investigate constraint satisfaction on inputs that are homomorphically equivalent to structures of bounded treewidth.
Constraint Satisfaction Problems And Finite Algebras
, 1999
"... Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NPcomplete in general, but certain restrictions on the form of the constraints can ensure tractability. In this paper we show that any restricted set of constraint types c ..."
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Cited by 64 (9 self)
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Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NPcomplete in general, but certain restrictions on the form of the constraints can ensure tractability. In this paper we show that any restricted set of constraint types can be associated with a finite universal algebra. We explore how the computational complexity of a restricted constraint satisfaction problem is connected to properties of the corresponding algebra. For this, we introduce a notion of `tractable algebra' and study how the tractability of an algebra relates to the tractability of its smaller derived algebras, including its subalgebras and homomorphic images. This allows us to significantly reduce the types of algebras which need to be investigated. Using these results we exhibit a common structural property of all known intractable constraint satisfaction problems. Finally, we classify all finite strictly simple surjective algebras wit...
Constraint satisfaction problems of bounded width
 IN: PROCEEDINGS OF FOCS 2009
, 2009
"... We provide a full characterization of applicability of The Local Consistency Checking algorithm to solving the nonuniform Constraint Satisfaction Problems. This settles the conjecture of Larose and Zádori. ..."
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Cited by 64 (6 self)
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We provide a full characterization of applicability of The Local Consistency Checking algorithm to solving the nonuniform Constraint Satisfaction Problems. This settles the conjecture of Larose and Zádori.