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Algebraic structures in combinatorial problems
 TECHNICAL REPORT, TECHNISCHE UNIVERSITAT DRESDEN
, 2001
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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 66 (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.
Towards a Dichotomy Theorem for the Counting Constraint Satisfaction Problem
, 2006
"... The Counting Constraint Satisfaction Problem (#CSP) can be expressed as follows: given a set of variables, a set of values that can be taken by the variables, and a set of constraints specifying some restrictions on the values that can be taken simultaneously by some variables, determine the number ..."
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Cited by 51 (9 self)
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The Counting Constraint Satisfaction Problem (#CSP) can be expressed as follows: given a set of variables, a set of values that can be taken by the variables, and a set of constraints specifying some restrictions on the values that can be taken simultaneously by some variables, determine the number of assignments of values to variables that satisfy all the constraints. The #CSP provides a general framework for numerous counting combinatorial problems including counting satisfying assignments to a propositional formula, counting graph homomorphisms, graph reliability and many others. This problem can be parametrized by the set of relations that may appear in a constraint. In this paper we start a systematic study of subclasses of the #CSP restricted in this way. The ultimate goal of this investigation is to distinguish those restricted subclasses of the #CSP which are solvable in polynomial time from those which are not. We show that the complexity of any restricted #CSP class on a finite domain can be deduced from the properties of polymorphisms of the allowed constraints, similar to that for the decision constraint satisfaction problem. Then we prove that if a subclass of the #CSP is solvable in polynomial time, then constraints allowed by the class satisfy some very restrictive condition: they need to have a Mal’tsev polymorphism, that is a ternary operation m(x, y, z) such that m(x, y, y) = m(y, y, x) = x. This condition uniformly explains many existing complexity results for particular cases of the #CSP, including the dichotomy results for the problem of counting graph homomorphisms, and it allows us to obtain new results.
The complexity of the counting constraint satisfaction problem
 In ICALP (1
, 2008
"... The Counting Constraint Satisfaction Problem (#CSP(H)) over a finite relational structureH can be expressed as follows: given a relational structure G over the same vocabulary, determine the number of homomorphisms from G toH. In this paper we characterize relational structuresH for which#CSP(H) can ..."
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Cited by 45 (7 self)
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The Counting Constraint Satisfaction Problem (#CSP(H)) over a finite relational structureH can be expressed as follows: given a relational structure G over the same vocabulary, determine the number of homomorphisms from G toH. In this paper we characterize relational structuresH for which#CSP(H) can be solved in polynomial time and prove that for all other structures the problem is #Pcomplete. 1
CSP dichotomy holds for digraphs with no sources and no sinks (a positive answer to the conjecture of BangJensen and Hell)
"... ... of graph homomorphisms) a CSP dichotomy for digraphs with no sources or sinks. The conjecture states that the constraint satisfaction problem for such a digraph is tractable if each component of its core is a circle and is NPcomplete otherwise. In this paper we prove this conjecture, and, as a ..."
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... of graph homomorphisms) a CSP dichotomy for digraphs with no sources or sinks. The conjecture states that the constraint satisfaction problem for such a digraph is tractable if each component of its core is a circle and is NPcomplete otherwise. In this paper we prove this conjecture, and, as a consequence, a conjecture of BangJensen, Hell and MacGillivray from 1995 classifying hereditarily hard digraphs. Further, we show that the CSP dichotomy for digraphs with no sources or sinks agrees with the algebraic characterization conjectured by Bulatov, Jeavons and Krokhin in 2005.
Universal algebra and hardness results for constraint satisfaction problems
, 2007
"... Abstract. We present algebraic conditions on constraint languages Γ that ensure the hardness of the constraint satisfaction problem CSP(Γ) for complexity classes L, NL, P, NP and ModpL. These criteria also give nonexpressibility results for various restrictions of Datalog. Furthermore, we show tha ..."
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Cited by 36 (7 self)
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Abstract. We present algebraic conditions on constraint languages Γ that ensure the hardness of the constraint satisfaction problem CSP(Γ) for complexity classes L, NL, P, NP and ModpL. These criteria also give nonexpressibility results for various restrictions of Datalog. Furthermore, we show that if CSP(Γ) is not firstorder definable then it is Lhard. Our proofs rely on tame congruence theory and on a finegrain analysis of the complexity of reductions used in the algebraic study of CSP. The results pave the way for a refinement of the dichotomy conjecture stating that each CSP(Γ) lies in P or is NPcomplete and they match the recent classification of [2] for Boolean CSP. We also infer a partial classification theorem for the complexity of CSP(Γ) when the associated algebra of Γ is the full idempotent reduct of a preprimal algebra. Constraint satisfaction problems (CSP) provide a unifying framework to study various computational problems arising naturally in artificial intelligence, combinatorial optimization, graph homomorphisms and database theory. An in
A characterisation of firstorder constraint satisfaction problems
 LOGICAL METHODS COMPUT. SCI
, 2007
"... We describe simple algebraic and combinatorial characterisations of finite relational core structures admitting finitely many obstructions. As a consequence, we show that it is decidable to determine whether a constraint satisfaction problem is firstorder definable: we show the general problem to ..."
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Cited by 34 (11 self)
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We describe simple algebraic and combinatorial characterisations of finite relational core structures admitting finitely many obstructions. As a consequence, we show that it is decidable to determine whether a constraint satisfaction problem is firstorder definable: we show the general problem to be NPcomplete, and give a polynomialtime algorithm in the case of cores. A slight modification of this algorithm provides, for firstorder definable CSP’s, a simple polytime algorithm to produce a solution when one exists. As an application of our algebraic characterisation of first order CSP’s, we describe a large family of Lcomplete CSP’s.
Generalized majorityminority operations are tractable
 In LICS
, 2005
"... Vol. 2 (4:1) 2006, pp. 1–15 www.lmcsonline.org ..."
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An Algebraic Approach To MultiSorted Constraints
 Proceedings of 9th International Conference on Principles and Practice of Constraint Programming
, 2003
"... We describe a common framework for the Constraint Satisfaction Problem and the Conjunctive Query Evaluation Problem, encompassing a generalised form of these problems in which different variables may take values from different sets. The framework we develop allows us to specify natural subclasses of ..."
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Cited by 23 (7 self)
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We describe a common framework for the Constraint Satisfaction Problem and the Conjunctive Query Evaluation Problem, encompassing a generalised form of these problems in which different variables may take values from different sets. The framework we develop allows us to specify natural subclasses of these two problems using algebraic techniques, and to establish when these subclasses are tractable. We show that a range of tractable classes can be obtained by combining recently identified tractable subclasses of the usual constraint satisfaction problem over a single set of values. We also systematically develop an algebraic structural theory for the general problem, which provides the prerequisites for further use of the powerful algebraic machinery.
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.