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Boolean Constraint Satisfaction Problems: When Does Post’s Lattice Help?

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Complexity of Constraints

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5250))

Satisfiability Problems

The propositional satisfiability problem SAT, i.e., the problem to decide, given a propositional formula φ (without loss of generality in conjunctive normal form CNF), if there is an assignment to the variables in φ that satisfies φ, is the historically first and standard NP-complete problem [Coo71]. However, there are well-known syntactic restrictions for which satisfiability is efficiently decidable, for example if every clause in the CNF formula has at most two literals (2CNF formulas) or if every clause has at most one positive literal (Horn formulas) or at most one negative literal (dual Horn formulas), see [KL99]. To study this phenomenon more generally, we study formulas with “clauses” of arbitrary shapes, i.e., consisting of applying arbitrary relations R ⊆ {0,1}k to (not necessarily distinct) variables x 1,...,x k . A constraint language Γ is a finite set of such relations. In the rest of this chapter, Γ and Γ′ will always denote Boolean constraint languages. A Γ-formula is a conjunction of clauses R(x 1,...,x k ) as above using only relations R from Γ. The for us central family of algorithmic problems, parameterized by a constraint language Γ, now is the problem to determine satisfiability of a given Γ-formula, denoted by Csp(Γ).

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Authors and Affiliations

  1. LIF (CNRS UMR 6166), Aix-Marseille Université, 163 avenue de Luminy, F-13288, Marseille, France

    Nadia Creignou

  2. Institut für Theoretische Informatik, Leibniz Universität Hannover, Appelstr. 4, D-30167, Hannover, Germany

    Heribert Vollmer

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  1. Nadia Creignou
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Editors and Affiliations

  1. LIF (CNRS UMR 6166), Aix-Marseille Université, 163 avenue de Luminy, F-13288, Marseille, France

    Nadia Creignou

  2. IBM Almaden Research Center, Computer Science Principles and Methodologies, and Computer Science Department, University of California, Santa Cruz, USA

    Phokion G. Kolaitis

  3. Institut für Theoretische Informatik, Leibniz, Universität Hannover, Appelstr. 4, 30167, Hannover, Germany

    Heribert Vollmer

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Creignou, N., Vollmer, H. (2008). Boolean Constraint Satisfaction Problems: When Does Post’s Lattice Help?. In: Creignou, N., Kolaitis, P.G., Vollmer, H. (eds) Complexity of Constraints. Lecture Notes in Computer Science, vol 5250. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92800-3_2

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