Counting Predicates, Subset Surjective Functions, and Counting CSPs

Bulatov, Andreǐ A.; Hedayaty, Amir

doi:10.1109/ismvl.2012.46

Cited by 4 publications

(2 citation statements)

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“…Since we initiated our work on constraint satisfaction with counting quantifiers, a possible algebraic approach has been published in [5,6]. It is clear reading our expositions that the combinatorics associated with our counting quantifiers is complex, and unfortunately the same seems to be the case on the algebraic side (where the relevant "expanding" polymorphisms have not previously been studied in their own right).…”

Section: Final Remarksmentioning

confidence: 99%

Constraint Satisfaction with Counting Quantifiers

Madelaine

Martin

Stacho

2012

Computer Science – Theory and Applications

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Section: Final Remarksmentioning

confidence: 99%

Constraint Satisfaction with Counting Quantifiers

Madelaine

Martin

Stacho

2012

Computer Science – Theory and Applications

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“…Finally, we give a description of Boolean max-co-clones, that is, sets of relations on {0, 1} closed under max-implementations. This is an extended version of [12].…”

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Galois correspondence for counting quantifiers

Bulatov,

Hedayaty

2012

Preprint

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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 preserving 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 be 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 max-quantification, 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. Finally, we give a description of Boolean max-co-clones, that is, sets of relations on {0, 1} closed under max-implementations. This is an extended version of [12].

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Constraint Satisfaction with Counting Quantifiers 2

Martin¹,

Stacho²

2014

Computer Science - Theory and Applications

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We study constraint satisfaction problems (CSPs) in the presence of counting quantifiers ∃ ≥j , asserting the existence of j distinct witnesses for the variable in question. As a continuation of our previous (CSR 2012) paper [11], we focus on the complexity of undirected graph templates. As our main contribution, we settle the two principal open questions proposed in [11]. Firstly, we complete the classification of clique templates by proving a full trichotomy for all possible combinations of counting quantifiers and clique sizes, placing each case either in P, NP-complete or Pspace-complete. This involves resolution of the cases in which we have the single quantifier ∃ ≥j on the clique K 2j . Secondly, we confirm a conjecture from [11], which proposes a full dichotomy for ∃ and ∃ ≥2 on all finite undirected graphs.The main thrust of this second result is the solution of the complexity for the infinite path which we prove is a polynomial-time solvable problem. By adapting the algorithm for the infinite path we are then able to solve the problem for finite paths, and then trees and forests. Thus as a corollary to this work, combining with the other cases from [11], we obtain a full dichotomy for ∃ and ∃ ≥2 quantifiers on finite graphs, each such problem being either in P or NP-hard. Finally, we persevere with the work of [11] in exploring cases in which there is dichotomy between P and Pspace-complete, in contrast with situations in which the intermediate NP-completeness may appear.Precisely the cases {j}-CSP(K 2j ) are left open here. Of course, {1}-CSP(K 2 ) is graph 2-colorability and is in L, but for j > 1 the situation was very unclear, and the referees noted specifically this lacuna.In this paper we settle this question, and find the surprising situation that {2}-The algorithm for the case {2}-CSP(K 4 ) is specialized and non-trivial, and consists in iteratively constructing a collection of forcing triples where we proceed to look for a contradiction.As a second focus of the paper, we continue the study of {1, 2}-CSP(H). In particular, we focus on finite undirected graphs for which a dichotomy was proposed in [11]. As a fundamental step towards this, we first investigate the complexity of {1, 2}-CSP(P ∞ ), where P ∞ denotes the infinite undirected path. We find tractability here in describing a particular unique obstruction, which takes the form of a special walk, whose presence or absence yields the answer to the problem. Again the algorithm is specialized and non-trivial, and in carefully augmenting it, we construct another polynomial-time algorithm, this time for all finite paths.Theorem 2. {1, 2}-CSP(P n ) is in P, for each n ∈ N.A corollary of this is the following key result. Corollary 3. {1, 2}-CSP(H) is in P, for each forest H.Combined with the results from [9,11], this allows us to observe a dichotomy for {1, 2}-CSP(H) as H ranges over undirected graphs, each problem being either in P or NP-hard, in turn settling a conjecture proposed in [11].In [11], the main preoccupation was in the distinction be...

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Counting Predicates, Subset Surjective Functions, and Counting CSPs

Cited by 4 publications

References 20 publications

Constraint Satisfaction with Counting Quantifiers

Constraint Satisfaction with Counting Quantifiers

Galois correspondence for counting quantifiers

Constraint Satisfaction with Counting Quantifiers 2

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