Structure identification of Boolean relations and plain bases for co-clones

Creignou, Nadia; Kolaitis, Phokion G.; Zanuttini, Bruno

doi:10.1016/j.jcss.2008.02.005

Cited by 40 publications

(71 citation statements)

References 12 publications

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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: 93%

“…In general, the problem of checking whether a κ-ary relation R is a base of a Boolean co-clone can be checked in time O(κ 2 |R|) using the algorithm in Creignou et al [CKZ08], and through exhaustive search, i.e. by repeatedly removing redundant columns and tuples, one can verify that the bases are also minimal.…”

Section: Lemma 38 the Bases For Irmentioning

confidence: 99%

“…Minimal Weak Bases of Boolean Co-Clones removing the second or third row gives a relation in IM 2 , and removing the fourth row gives a relation in IS 2 10 . These properties can efficiently be tested using the algorithm in Creignou et al [CKZ08]. Hence, there is no relation R ⊂ R IE 2 such that {R } = IE 2 from which it follows that R IE 2 is a minimal weak base.…”

Section: Lemma 310 the Bases Formentioning

confidence: 99%

“…Hence, if R IS κ 00 ∈ {R} ∃ then R IS κ 00 ∈ S ∃ . In the rest of this proof, let λ > κ denote the arity of R. We must prove that R IS κ 00 ∈ {R} ∃ , and by Creignou et al [CKZ08] we know that R can be expressed as an IHSB κ + formula ϕ(y 1 , . .…”

Section: Lemma 310 the Bases Formentioning

confidence: 99%

“…Our proofs are based on comparing the least expressive language in the co-clone with the most expressive language in the co-clone, in order to obtain an upper bound of p. The least expressive language in a co-clone in this context is nothing else than a weak base of the co-clone, as studied in the previous chapter. The most expressive language in a co-clone is called a plain base, and were introduces by Creignou et al [CKZ08]. We first give a general result and provide a sufficient condition for a co-clone over any finite domain to be polynomially closed: a co-clone Inv(C) is polynomially closed if C contains a k-ary near-unanimity function for some k ≥ 3.…”

Section: Introductionmentioning

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: 93%

Section: Lemma 38 the Bases For Irmentioning

confidence: 99%

Section: Lemma 310 the Bases Formentioning

confidence: 99%

Section: Lemma 310 the Bases Formentioning

confidence: 99%

Section: Introductionmentioning

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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The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits

Schwerdtfeger

2014

Computer Science - Theory and Applications

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For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. In 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs [GKMP09], motivated mainly by research on satisfiability algorithms and the satisfiability threshold. They proved dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question. Recently, we were able to establish the trichotomy [Sch13].Here, we consider connectivity issues of satisfiability problems defined by Boolean circuits and propositional formulas that use gates, resp. connectives, from a fixed set of Boolean functions. We obtain dichotomies for the diameter and the two connectivity problems: On one side, the diameter is linear in the number of variables, and both problems are in P, while on the other side, the diameter can be exponential, and the problems are PSPACE-complete. For partially quantified formulas, we show an analogous dichotomy.

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Non-uniform Boolean Constraint Satisfaction Problems with Cardinality Constraint

Creignou

Schnoor

2008

Computer Science Logic

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Structure identification of Boolean relations and plain bases for co-clones

Cited by 40 publications

References 12 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 Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits

Non-uniform Boolean Constraint Satisfaction Problems with Cardinality Constraint

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