Bases for Boolean co-clones

Böhler, Elmar; Reith, Steffen; Schnoor, Henning; Vollmer, Heribert

doi:10.1016/j.ipl.2005.06.003

Cited by 51 publications

(82 citation statements)

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“…First, each weak base that we determine can in a natural sense be considered to be minimal. Second, we present alternative proofs where Schnoor's and Schnoor's procedure results in relations which are exponentially larger than the bases given by Böhler et al [BSRV05] and Creignou et al [CKZ08]. Third, we extend the notion of a weak base of a co-clone Inv(C) to also cover the case where Inv(C) is of infinite order.…”

Section: Introductionmentioning

confidence: 94%

“…This is not due to lack of imagination on the authors part, but because there is a deeper relationship between clones and co-clones, which we investigate in the forthcoming section. Bases for Boolean co-clones were first investigated by Böhler et al [BSRV05]. We do not include this list since we in the later sections are interested in bases fulfilling some additional properties.…”

Section: Clone Theorymentioning

confidence: 99%

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Strong Partial Clones and the Complexity of Constraint Satisfaction Problems : Limitations and Applications

Lagerkvist¹

2015

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In this thesis we study the worst-case time complexity of the constraint satisfaction problem parameterized by a constraint language (CSP(S)), which is the problem of determining whether a conjunctive formula over S has a model. To study the complexity of CSP(S) we borrow methods from universal algebra. In particular, we consider algebras of partial functions, called strong partial clones. This algebraic approach allows us to obtain a more nuanced view of the complexity CSP(S) than possible with algebras of total functions, clones.The results of this thesis is split into two main parts. In the first part we investigate properties of strong partial clones, beginning with a classification of weak bases for all Boolean relational clones. Weak bases are constraint languages where the corresponding strong partial clones in a certain sense are extraordinarily large, and they provide a rich amount of information regarding the complexity of the corresponding CSP problems. We then proceed by classifying the Boolean relational clones according to whether it is possible to represent every relation by a conjunctive, logical formula over the weak base without needing more than a polynomial number of existentially quantified variables. A relational clone satisfying this condition is called polynomially closed and we show that this property has a close relationship with the concept of few subpowers. Using this classification we prove that a strong partial clone is of infinite order if (1) the total functions in the strong partial clone are essentially unary and (2) the corresponding constraint language is finite. Despite this, we prove that these strong partial clones can be succinctly represented with finite sets of partial functions, bounded bases, by considering stronger notions of closure than functional composition.In the second part of this thesis we apply the theory developed in the first part. We begin by studying the complexity of CSP(S) where S is a Boolean constraint language, the generalised satisfiability problem (SAT(S)). Using weak bases we prove that there exists a relation R = = = 1/3 such that SAT({R = = = 1/3 }) is the easiest NP-complete SAT(S) problem. We rule out the possibility that SAT({R = = = 1/3 }) is solvable in subexponential time unless a well-known complexity theoretical conjecture, the exponential-time hypothesis, (ETH) is false. We then proceed to study the computational complexity of two optimisation variants of the SAT(S) problem: the maximum ones problem over a Boolean constraint language S (MAX-ONES(S)) and the valued constraint satisfaction problem over a set of Boolean cost functions ∆ (VCSP(∆)). For MAX-ONES(S) we use partial clone theory and prove that MAX-ONES({R = = = 1/3 }) is the easiest NPcomplete MAX-ONES(S) problem. These algebraic techniques do not work for VCSP(∆), however, where we instead use multimorphisms to prove that MAX-CUT is the easiest NP-complete Boolean VCSP(∆) problem. Similar to the case of SAT(S) we then rule out the possibility of subexponential algorith...

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Section: Introductionmentioning

confidence: 94%

Section: Clone Theorymentioning

confidence: 99%

Strong Partial Clones and the Complexity of Constraint Satisfaction Problems : Limitations and Applications

Lagerkvist¹

2015

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“…This special case has been studied in detail in [30]. Here, we give a brief independent derivation of the relevant results using the techniques introduced above.…”

Section: Proposition 32 For Allmentioning

confidence: 99%

The Complexity of Valued Constraint Satisfaction Problems

Živný¹

2012

Cognitive Technologies

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“…As for the above introduced classes, IHSB relations can be characterized by their polymorphism, as follows immediately from [BRSV05]:…”

Section: Fig 2 List Of All Boolean Clones With Their Defining Basesmentioning

confidence: 99%

The Complexity of Problems for Quantified Constraints

et al. 2009

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Abstract. In this paper we will look at restricted versions of the evaluation problem, the model checking problem, the equivalence problem, and the counting problem for quantified propositional formulas, both with and without bound on the number of quantifier alternations. The restrictions are such that we consider formulas in conjunctive normal-form with restricted types of clauses (e.g., positive, Horn, linear, etc.). For each of these algorithmic goals we will obtain full complexity classifications, exhibiting on the one hand severe syntactic restrictions of the original problems that are still computationally hard, and on the other hand non-trivial subcases that admit efficient solution algorithms. Generalizing these results to non Boolean domains, we obtain a number of hardnes results for quantified constraints over arbitrary finite universes.

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Bases for Boolean co-clones

Cited by 51 publications

References 4 publications

Strong Partial Clones and the Complexity of Constraint Satisfaction Problems : Limitations and Applications

Strong Partial Clones and the Complexity of Constraint Satisfaction Problems : Limitations and Applications

The Complexity of Valued Constraint Satisfaction Problems

The Complexity of Problems for Quantified Constraints

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