The complexity of satisfiability problems: Refining Schaefer's theorem

Allender, Eric; Bauland, Michael; Immerman, Neil; Schnoor, Henning; Vollmer, Heribert

doi:10.1016/j.jcss.2008.11.001

Cited by 35 publications

(74 citation statements)

References 25 publications

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“…In general the problem is NP-complete, but restricting the type of constraint relations involved may yield tractable problems. In fact, Schaefer [Sch78] and more recently [ABISV05] have completely classified the complexity of Boolean CSP's and from their work it follows that Boolean CSP's are either trivial, first-order definable, or complete (under AC 0 reductions) for one of the following standard classes of problems: L, NL, P, ⊕L and NP. One of the outstanding problems in the field is the so-called dichotomy conjecture [FV93] that states that every CSP should be either in P or NP-complete.…”

Section: Introductionmentioning

confidence: 99%

A Characterisation of First-Order Constraint Satisfaction Problems

Larose¹,

Loten²,

Tardif³

2007

Logical Methods in Computer Science

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Abstract. We describe simple algebraic and combinatorial characterisations of finite relational core structures admitting finitely many obstructions. As a consequence, we show that it is decidable to determine whether a constraint satisfaction problem is first-order definable: we show the general problem to be NP-complete, and give a polynomial-time algorithm in the case of cores. A slight modification of this algorithm provides, for firstorder definable CSP's, a simple poly-time algorithm to produce a solution when one exists. As an application of our algebraic characterisation of first order CSP's, we describe a large family of L-complete CSP's.

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Section: Introductionmentioning

confidence: 99%

A Characterisation of First-Order Constraint Satisfaction Problems

Larose¹,

Loten²,

Tardif³

2007

Logical Methods in Computer Science

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“…The conditions of "tractability" -here provability over RCA 0 -differ from those of Schaefer dichotomy theorem but the considered classes of relations remain the same. We obtain the surprising result that infinite versions of horn and co-horn satisfaction problems are provable over RCA 0 and strictly weaker than bijunctive and affine corresponding principles, whereas the complexity classification of [1] has shown that horn satisfiability was P-complete under AC 0 reduction, hence at least as strong as bijunctive satisfiability which is NL-complete.…”

Section: Conclusion and Questionsmentioning

confidence: 84%

“…Many other dichotomy theorems have been proven since, about refinements to AC 0 reductions [1], variants about counting, optimization, 3-valued domains and many others [4,7,3]. The existence of dichotomies for n-valued domains with n > 3 remains open.…”

Section: Introductionmentioning

confidence: 99%

The Complexity of Satisfaction Problems in Reverse Mathematics

Patey

2014

Language, Life, Limits

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Abstract. Satisfiability problems play a central role in computer science and engineering as a general framework for studying the complexity of various problems. Schaefer proved in 1978 that truth satisfaction of propositional formulas given a language of relations is either NP-complete or tractable. We classify the corresponding satisfying assignment construction problems in the framework of reverse mathematics and show that the principles are either provable over RCA 0 or equivalent to WKL 0 . We formulate also a Ramseyan version of the problems and state a different dichotomy theorem. However, the different classes arising from this classification are not known to be distinct.

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“…To date only four complexity classes and a series of very similar classes inside P are known such that CSP(Γ) can be complete in [2,29]. In some cases the lack of problems CSP(Γ) of intermediate complexity is shown [29].…”

Section: Independentmentioning

confidence: 99%

Constraint satisfaction problems and global cardinality constraints

Bulatov

Marx

2010

Commun. ACM

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In a constraint satisfaction problem (CSP) the goal is to find an assignment of a given set of variables subject to specified constraints. A global cardinality constraint is an additional requirement that prescribes how many variables must be assigned a certain value. We study the complexity of the problem CCSP(Γ), the constraint satisfaction problem with global cardinality constraints that allows only relations from the set Γ. The main result of this paper characterizes sets Γ that give rise to problems solvable in polynomial time, and states that the remaining such problems are NP-complete. CONSTRAINT PROBLEMS Constraint Satisfaction ProblemAmong formalisms unifying and classifying various combinatorial problems the Constraint Satisfaction Problem (or CSP) is one of the most successful ones. In this problem we are given a set of variables and a collection of restrictionsconstraints -on the allowed combinations of values of the variables; the goal is to find an assignment to the variables so that all constraints are satisfied. Usually constraints are imposed on small sets of variables, thus, the CSP formalizes the idea of finding a global solution bound by local restrictions. The Sudoku puzzle gives a popular toy example of CSP. We need to assign values -numbers from 1 to 9 -to variables -entries of the puzzle so that the values of variables in a row, column, or 3 × 3 block are different. Another toy example whose CSP encoding is less obvious is the 8-Queen problem: place 8 queens on a 8 × 8 chessboard so that they do not hit each other [15]. To represent it as a CSP we consider the columns {a, b, c, d, e, f, g, h} (see Fig. 1) as variables that can be assigned values from the set of rows, and the assigned value shows the position of a queen in this column.Many combinatorial problems readily fall into this framework. For example, in the Graph 3-Coloring problem the vertices of a given graph are variables to receive one of the three colors, and assignments are constrained by the requirement that adjacent vertices receive different colors. Thus, this problem is a CSP. The list of examples can be extended Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Copyright 2008 ACM 0001-0782/08/0X00 ...$5.00. Figure 1: The 8-Queens problemby other combinatorial problems like Satisfiability, problems in scheduling, temporal and spatial reasoning, and many others.Constraint satisfaction problems have been studied from both practical and theoretical perspectives. On the practical side the expressive power of the CSP allows to model a wide range of real-world problems from planning [24] and scheduling [35], frequency assignment problems [17], to image processing ...

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The complexity of satisfiability problems: Refining Schaefer's theorem

Cited by 35 publications

References 25 publications

A Characterisation of First-Order Constraint Satisfaction Problems

A Characterisation of First-Order Constraint Satisfaction Problems

The Complexity of Satisfaction Problems in Reverse Mathematics

Constraint satisfaction problems and global cardinality constraints

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