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Tractability and learnability arising from algebras with few subpowers
 In LICS’07
, 2007
"... A kedge operation ϕ on a finite set A is a k + 1ary operation that satisfies the identities ..."
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Cited by 53 (7 self)
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A kedge operation ϕ on a finite set A is a k + 1ary operation that satisfies the identities
The Complexity of Soft Constraint Satisfaction
, 2006
"... Over the past few years there has been considerable progress in methods to systematically analyse the complexity of constraint satisfaction problems with specified constraint types. One very powerful theoretical development in this area links the complexity of a set of constraints to a corresponding ..."
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Cited by 44 (13 self)
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Over the past few years there has been considerable progress in methods to systematically analyse the complexity of constraint satisfaction problems with specified constraint types. One very powerful theoretical development in this area links the complexity of a set of constraints to a corresponding set of algebraic operations, known as polymorphisms. In this paper we extend the analysis of complexity to the more general framework of combinatorial optimisation problems expressed using various forms of soft constraints. We launch a systematic investigation of the complexity of these problems by extending the notion of a polymorphism to a more general algebraic operation, which we call a multimorphism. We show that many tractable sets of soft constraints, both established and novel, can be characterised by the presence of particular multimorphisms. We also show that a simple set of NPhard constraints has very restricted multimorphisms. Finally, we use the notion of multimorphism to give a complete classification of complexity for the Boolean case which extends several earlier classification results for particular special cases.
The computational complexity of quantified constraint satisfaction
, 2004
"... The constraint satisfaction problem (CSP) is a framework for modelling search problems. An instance of the CSP consists of a set of variables and a set of constraints on the variables; the question is to decide whether or not there is an assignment to the variables satisfying all of the constraints. ..."
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The constraint satisfaction problem (CSP) is a framework for modelling search problems. An instance of the CSP consists of a set of variables and a set of constraints on the variables; the question is to decide whether or not there is an assignment to the variables satisfying all of the constraints. The quantified constraint satisfaction problem (QCSP) is a generalization of the CSP in which variables may be both universally and existentially quantified. The general intractability of the CSP and QCSP motivates the search for restricted cases of these problems that are polynomialtime tractable. In this
Varieties with few subalgebras of powers
, 2006
"... Abstract. The Constraint Satisfaction Problem Dichotomy Conjecture of Feder and Vardi [12] has in the last 10 years been profitably reformulated as a conjecture about the set SPfin(A) of subalgebras of finite Cartesian powers of a finite universal algebra A [20, 5]. One particular strategy, advanced ..."
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Abstract. The Constraint Satisfaction Problem Dichotomy Conjecture of Feder and Vardi [12] has in the last 10 years been profitably reformulated as a conjecture about the set SPfin(A) of subalgebras of finite Cartesian powers of a finite universal algebra A [20, 5]. One particular strategy, advanced by Dalmau in his doctoral thesis [8], has confirmed the conjecture for a certain class of finite algebras A which, among other things, have the property that the number of subalgebras of An is bounded by an exponential polynomial. In this paper we characterize the finite algebras A with this property, which we call having few subpowers, and develop a representation theory for the subpowers of algebras having few subpowers. Our characterization shows that algebras having few subpowers are the finite members of a newly discovered and surprisingly robust Maltsev class defined by the existence of a special term we call an edge term. We also prove some tight connections between the asymptotic behavior of the number of subalgebras of An and some related functions on the one hand, and some standard algebraic properties of A on the
Recent results on the algebraic approach to the CSP
 In The Same Volume
, 2008
"... Abstract. We describe an algebraic approach to the constraint satisfaction problem (CSP) and present recent results on the CSP that make use of, in an essential way, this algebraic framework. 1 ..."
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Abstract. We describe an algebraic approach to the constraint satisfaction problem (CSP) and present recent results on the CSP that make use of, in an essential way, this algebraic framework. 1
The structure of tractable constraint satisfaction problems
 In MFCS 2006
, 2006
"... Abstract We give a survey of recent results on the complexity of constraint satisfaction problems. Our main emphasis is on tractable structural restrictions. 1 ..."
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Abstract We give a survey of recent results on the complexity of constraint satisfaction problems. Our main emphasis is on tractable structural restrictions. 1
Colouring, constraint satisfaction, and complexity
"... Constraint satisfaction problems have enjoyed much attention since the early seventies, and in the last decade have become also a focus of attention amongst theoreticians. Graph colourings are a special class of constraint satisfaction problems; they offer a microcosm of many of the considerations t ..."
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Cited by 18 (1 self)
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Constraint satisfaction problems have enjoyed much attention since the early seventies, and in the last decade have become also a focus of attention amongst theoreticians. Graph colourings are a special class of constraint satisfaction problems; they offer a microcosm of many of the considerations that occur in constraint satisfaction. From the point of view of theory, they are well known to exhibit a dichotomy of complexity the kcolouring problem is polynomial time solvable when k ≤ 2, and NPcomplete when k ≥ 3. Similar dichotomy has been proved for the class of graph homomorphism problems, which are intermediate problems between graph colouring and constraint satisfaction
The Complexity of Constraint Satisfaction Games and QCSP
"... We study the complexity of twoperson constraint satisfaction games. An instance of such a game is given by a collection of constraints on overlapping sets of variables, and the two players alternately make moves assigning values from a finite domain to the variables in a specified order. The first ..."
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We study the complexity of twoperson constraint satisfaction games. An instance of such a game is given by a collection of constraints on overlapping sets of variables, and the two players alternately make moves assigning values from a finite domain to the variables in a specified order. The first player tries to satisfy all constraints, while the other tries to break at least one constraint; the goal is to decide whether the first player has a winning strategy. We show that such games can be conveniently represented by a logical form of quantified constraint satisfaction, where an instance is given by a firstorder sentence in which quantifiers alternate and the quantifierfree part is a conjunction of atomic formulas; the goal is to decide whether the sentence is true. While the problem of deciding such a game is PSPACEcomplete in general, by restricting the set of allowed constraint predicates, one can obtain infinite classes of constraint satisfaction games of lower complexity. We use the quantified constraint satisfaction framework to study how the complexity of deciding such a game depends on the parameter set of allowed predicates. With every predicate, one can associate certain predicatepreserving operations, called polymorphisms. We show that the complexity of our games is determined by the surjective polymorphisms of the constraint predicates. We illustrate how this result can be used by identifying the complexity of a wide variety of constraint satisfaction games.
The dichotomy for conservative constraint satisfaction problems revisited
 In Proceedings of the 26th Annual IEEE Symposium on Logic in Computer Science, LICS 2011
"... Abstract—A central open question in the study of nonuniform constraint satisfaction problems (CSPs) is the dichotomy conjecture of Feder and Vardi stating that the CSP over a fixed constraint language is either NPcomplete, or tractable. One of the main achievements in this direction is a result of ..."
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Abstract—A central open question in the study of nonuniform constraint satisfaction problems (CSPs) is the dichotomy conjecture of Feder and Vardi stating that the CSP over a fixed constraint language is either NPcomplete, or tractable. One of the main achievements in this direction is a result of Bulatov (LICS’03) confirming the dichotomy conjecture for conservative CSPs, that is, CSPs over constraint languages containing all unary relations. Unfortunately, the proof is very long and complicated, and therefore hard to understand even for a specialist. This paper provides a short and transparent proof.
On tractability and congruence distributivity
, 2007
"... Constraint languages that arise from finite algebras have recently been the object of study, especially in connection with the Dichotomy Conjecture of Feder and Vardi. An important class of algebras are those that generate congruence distributive varieties and included among this class are lattices, ..."
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Cited by 14 (1 self)
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Constraint languages that arise from finite algebras have recently been the object of study, especially in connection with the Dichotomy Conjecture of Feder and Vardi. An important class of algebras are those that generate congruence distributive varieties and included among this class are lattices, and more generally, those algebras that have nearunanimity term operations. An algebra will generate a congruence distributive variety if and only if it has a sequence of ternary term operations, called Jónsson terms, that satisfy certain equations. We prove that constraint languages consisting of relations that are invariant under a short sequence of Jónsson terms are tractable by showing that such languages have bounded relational width.