The complexity of the counting constraint satisfaction problem

Bulatov, Andreǐ A.

doi:10.1145/2528400

Cited by 91 publications

(21 citation statements)

References 60 publications

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“…Although it is not known if there is a dichotomy for CSP, Bulatov has recently shown that, for every Γ , #CSP(Γ ) is either computable in polynomial time or #P-complete [4]. Two of the present authors have since given an elementary proof of this result and also shown the dichotomy to be decidable [24].…”

Section: Counting Cspmentioning

confidence: 64%

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The complexity of approximating bounded-degree Boolean #CSP

Dyer

Goldberg

Jalsenius

et al. 2012

Information and Computation

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The degree of a CSP instance is the maximum number of times that any variable appears in the scopes of constraints. We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum allowed degree is large enough (at least 6) we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of {0}, {1} and binary implication. Otherwise, there is no FPRAS unless NP = RP. For lower degree bounds, additional cases arise, where the complexity is related to the complexity of approximately counting independent sets in hypergraphs.

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Section: Counting Cspmentioning

confidence: 64%

“…This should not be confused with the concept of primitive positive definability (pp-definability) which appears in algebraic treatments of CSP and #CSP, for example in the work of Bulatov[4].…”

mentioning

confidence: 99%

The complexity of approximating bounded-degree Boolean #CSP

Dyer

Goldberg

Jalsenius

et al. 2012

Information and Computation

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“…It is again the polymorphisms of a language which determine tractability of #CSP( ). Bulatov has characterised all languages for which #CSP( ) can be solved in polynomial time and proved that for all other languages the problem is #P-complete [26]. All tractable languages have a Mal'tsev polymorphism.…”

Section: Quantified Csp Uncertain Csp #Csp and Related Problemsmentioning

confidence: 99%

Tractability in constraint satisfaction problems: a survey

Carbonnel

Cooper

2015

Constraints

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Even though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP. What is tractability?The idea that an algorithm is efficient if its time complexity is a polynomial function of the size of its input can be traced back to pioneering work of Cobham [45] and Edmonds [83], but the foundations of complexity theory are based on the seminal work of Cook [52] and Karp [118]. A computational decision problem (such as the constraint satisfaction problem or CSP) consists of a generic instance (in the case of the CSP, a set of variables, their

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“…The complexity of the exact counting problem (when we are required to find the exact number of solutions) is settled in [6] by showing that for any finite D and any set Γ of relations over D the problem is polynomial time solvable or is complete in a natural complexity class #P . One of the key steps in that line of research is the following result: For a relation R and a set of relations Γ over D, such that R belongs to the co-clone generated by Γ, then #CSP(Γ ∪ {R}) is polynomial time reducible to #CSP(Γ).…”

Section: Max-quantifiersmentioning

confidence: 99%

“…For the decision problem, a number of very strong results have been proved using methods of universal algebra [1], [4], [5], [9], [15]. For the exact counting CSP a complete complexity classification of such problems has been obtained [6]. Substantial progress has been also made in the quantified CSP [10].…”

Section: Introductionmentioning

confidence: 99%

Counting Predicates, Subset Surjective Functions, and Counting CSPs

Bulatov

Hedayaty

2012

2012 IEEE 42nd International Symposium on Multiple-Valued Logic

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We introduce a new type of closure operator on the set of relations, max-implementation, and its weaker analog max-quantification. Then we show that approximation reductions between counting constraint satisfaction problems (CSPs) are preserved by these two types of closure operators. Together with some previous results this means that the approximation complexity of counting CSPs is determined by partial clones of relations that additionally closed under these new types of closure operators. Galois correspondence of various kind have proved to been quite helpful in the study of the complexity of the CSP. While we were unable to identify a Galois correspondence for partial clones closed under max-implementation and maxquantification, we obtain such results for slightly different type of closure operators, k-existential quantification. This type of quantifiers are known as counting quantifiers in model theory, and often used to enhance first order logic languages. We characterize partial clones of relations closed under k-existential quantification as sets of relations invariant under a set of partial functions that satisfy the condition of k-subset surjectivity.

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The complexity of the counting constraint satisfaction problem

Abstract: The Counting Constraint Satisfaction Problem (#CSP(H)) over a finite relational structure H can be expressed as follows: given a relational structure G over the same vocabulary, determine the number of homomorphisms from

Cited by 91 publications

References 60 publications

The complexity of approximating bounded-degree Boolean #CSP

The complexity of approximating bounded-degree Boolean #CSP

Tractability in constraint satisfaction problems: a survey

Counting Predicates, Subset Surjective Functions, and Counting CSPs

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