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36
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 35 (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.
Dualities for constraint satisfaction problems
"... In a nutshell, a duality for a constraint satisfaction problem equates the existence of one homomorphism to the nonexistence of other homomorphisms. In this survey paper, we give an overview of logical, combinatorial, and algebraic aspects of the following forms of duality for constraint satisfact ..."
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Cited by 22 (8 self)
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In a nutshell, a duality for a constraint satisfaction problem equates the existence of one homomorphism to the nonexistence of other homomorphisms. In this survey paper, we give an overview of logical, combinatorial, and algebraic aspects of the following forms of duality for constraint satisfaction problems: finite duality, bounded pathwidth duality, and bounded treewidth duality.
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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Cited by 21 (4 self)
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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
Symmetric Datalog and constraint satisfaction problems in Logspace
 IN LICS’07
, 2007
"... We introduce symmetric Datalog, a syntactic restriction of linear Datalog and show that its expressive power is exactly that of restricted symmetric monotone Krom SNP. The deep result of Reingold [17] on the complexity of undirected connectivity suffices to show that symmetric Datalog queries can be ..."
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Cited by 12 (7 self)
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We introduce symmetric Datalog, a syntactic restriction of linear Datalog and show that its expressive power is exactly that of restricted symmetric monotone Krom SNP. The deep result of Reingold [17] on the complexity of undirected connectivity suffices to show that symmetric Datalog queries can be evaluated in logarithmic space. We show that for a number of constraint languages Γ, the complement of the constraint satisfaction problem CSP(Γ) can be expressed in symmetric Datalog. In particular, we show that if CSP(Γ) is firstorder definable and Λ is a finite subset of the relational clone generated by Γ then ¬CSP(Λ) is definable in symmetric Datalog. Over the twoelement domain and under a standard complexitytheoretic assumption, expressibility of ¬CSP(Γ) in symmetric Datalog corresponds exactly to the class of CSPs solvable in logarithmic space. Finally, we describe a fairly general subclass of implicational (or 0/1/all) constraints for which the complement of the corresponding CSP is also definable in symmetric Datalog. Our results provide preliminary evidence that symmetric Datalog may be a unifying explanation for families of CSPs lying in L.
CSP dichotomy for special triads
 PROC. AMER. MATH. SOC
"... For a fixed digraph G, the Constraint Satisfaction Problem with the template G, or CSP(G) for short, is the problem of deciding whether a given input digraph H admits a homomorphism to G. The dichotomy conjecture of Feder and Vardi states that CSP(G), for any choice of G, is solvable in polynomial ..."
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Cited by 11 (2 self)
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For a fixed digraph G, the Constraint Satisfaction Problem with the template G, or CSP(G) for short, is the problem of deciding whether a given input digraph H admits a homomorphism to G. The dichotomy conjecture of Feder and Vardi states that CSP(G), for any choice of G, is solvable in polynomial time or NPcomplete. This paper confirms the conjecture for a class of oriented trees called special triads. As a corollary we get the smallest known example of an oriented tree (with 33 vertices) defining an NPcomplete CSP(G).
Caterpillar duality for constraint satisfaction problems
 In LICS’08
, 2008
"... The study of constraint satisfaction problems definable in various fragments of Datalog has recently gained considerable importance. We consider constraint satisfaction problems that are definable in the smallest natural recursive fragment of Datalog monadic linear Datalog with at most one EDB per ..."
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Cited by 6 (1 self)
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The study of constraint satisfaction problems definable in various fragments of Datalog has recently gained considerable importance. We consider constraint satisfaction problems that are definable in the smallest natural recursive fragment of Datalog monadic linear Datalog with at most one EDB per rule. We give combinatorial and algebraic characterisations of such problems, in terms of caterpillar dualities and lattice operations, respectively. We then apply our results to study graph Hcolouring problems. 1
Maltsev + datalog –> symmetric datalog
 In LICS
, 2008
"... Let B be a finite, core relational structure and let A be the algebra associated to B, i.e. whose terms are the operations on the universe of B that preserve the relations of B. We show that if A generates a socalled arithmetical variety then CSP(B), the constraint satisfaction problem associated ..."
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Cited by 6 (1 self)
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Let B be a finite, core relational structure and let A be the algebra associated to B, i.e. whose terms are the operations on the universe of B that preserve the relations of B. We show that if A generates a socalled arithmetical variety then CSP(B), the constraint satisfaction problem associated to B, is solvable in Logspace; in fact ¬CSP(B) is expressible in symmetric Datalog. In particular, we obtain that if ¬CSP(B) is expressible in Datalog and the relations of B are invariant under a Maltsev operation then ¬CSP(B) is in symmetric Datalog.
Relatively Quantified Constraint Satisfaction
"... The constraint satisfaction problem (CSP) is a convenient framework for modelling search problems; the CSP involves deciding, given a set of constraints on variables, whether or not there is an assignment to the variables satisfying all of the constraints. This paper is concerned with the more gener ..."
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Cited by 5 (2 self)
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The constraint satisfaction problem (CSP) is a convenient framework for modelling search problems; the CSP involves deciding, given a set of constraints on variables, whether or not there is an assignment to the variables satisfying all of the constraints. This paper is concerned with the more general framework of quantified constraint satisfaction, in which variables can be quantified both universally and existentially. We study the relatively quantified constraint satisfaction problem (RQCSP), in which the values for each individual variable can be arbitrarily restricted. We give a complete complexity classification of the cases of the RQCSP where the types of constraints that may appear are specified by a constraint language.
Algebraic Properties of Valued Constraint Satisfaction Problem
"... Abstract. The paper presents an algebraic framework for optimization problems expressible as Valued Constraint Satisfaction Problems. Our results generalize the algebraic framework for the decision version (CSPs) provided by Bulatov et al. [SICOMP 2005]. We introduce the notions of weighted algebras ..."
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Cited by 4 (0 self)
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Abstract. The paper presents an algebraic framework for optimization problems expressible as Valued Constraint Satisfaction Problems. Our results generalize the algebraic framework for the decision version (CSPs) provided by Bulatov et al. [SICOMP 2005]. We introduce the notions of weighted algebras and varieties and use the Galois connection due to Cohen et al. [SICOMP 2013] to link VCSP languages to weighted algebras. We show that the difficulty of VCSP depends only on the weighted variety generated by the associated weighted algebra. Paralleling the results for CSPs we exhibit a reduction to cores and rigid cores which allows us to focus on idempotent weighted varieties. Further, we propose an analogue of the Algebraic CSP Dichotomy Conjecture; prove the hardness direction and verify that it agrees with known results for VCSPs on twoelement sets [Cohen et al. 2006], finitevalued VCSPs [Thapper and Živny ́ 2013] and conservative VCSPs [Kolmogorov and Živny ́ 2013]. 1
Path homomorphisms, graph colourings and boolean matrices
"... We investigate bounds on the chromatic number of a graph G derived from the nonexistence of homomorphisms from some path ~P into some orientation ~G of G. The condition is often efficiently verifiable using boolean matrix multiplications. However, the bound associated to a path ~P depends on the re ..."
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Cited by 3 (2 self)
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We investigate bounds on the chromatic number of a graph G derived from the nonexistence of homomorphisms from some path ~P into some orientation ~G of G. The condition is often efficiently verifiable using boolean matrix multiplications. However, the bound associated to a path ~P depends on the relation between the “algebraic length” and “derived algebraic length ” of ~P. This suggests that paths yielding efficient bounds may be exponentially large with respect to G, and the corresponding heuristic may not be constructive. ∗Partially supported by the Project LN00A056 of the Czech Ministery of Education and by CRM Barcelona, Spain. †Supported by grants from NSERC and ARP 1 1