Classifying the Complexity of Constraints Using Finite Algebras
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It is shown that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra is explored.Abstract:
Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. Here we show that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and we explore how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra. Hence, we completely translate the problem of classifying the complexity of restricted constraint satisfaction problems into the language of universal algebra.
We introduce a notion of "tractable algebra," and investigate how the tractability of an algebra relates to the tractability of the smaller algebras which may be derived from it, including its subalgebras and homomorphic images. This allows us to reduce significantly the types of algebras which need to be classified. Using our results we also show that if the decision problem associated with a given collection of constraint types can be solved efficiently, then so can the corresponding search problem. We then classify all finite strictly simple surjective algebras with respect to tractability, obtaining a dichotomy theorem which generalizes Schaefer's dichotomy for the generalized satisfiability problem. Finally, we suggest a possible general algebraic criterion for distinguishing the tractable and intractable cases of the constraint satisfaction problem.read more
Citations
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Book
Handbook of Constraint Programming
TL;DR: Researchers from other fields should find in this handbook an effective way to learn about constraint programming and to possibly use some of the constraint programming concepts and techniques in their work, thus providing a means for a fruitful cross-fertilization among different research areas.
Journal ArticleDOI
A dichotomy theorem for constraint satisfaction problems on a 3-element set
TL;DR: Every subproblem of the CSP is either tractable or NP-complete, and the criterion separating them is that conjectured in Bulatov et al.
Journal ArticleDOI
A Linear Programming Approach to Max-Sum Problem: A Review
TL;DR: This work reviews a not widely known approach to the max-sum labeling problem, developed by Ukrainian researchers Schlesinger et al. in 1976, and shows how it contributes to recent results, most importantly, those on the convex combination of trees and tree-reweighted max-product.
Proceedings ArticleDOI
A Dichotomy Theorem for Nonuniform CSPs
TL;DR: In this paper, the authors confirm the Dichotomy Conjecture for the non-uniform CSP, which states that for every constraint language \Gm the problem is either solvable in polynomial time or is NP-complete.
Posted Content
A dichotomy theorem for nonuniform CSPs
TL;DR: In this article, the Dichotomy Conjecture on the complexity of non-uniform constraint satisfaction problems was shown to hold for the special case of constraint satisfaction with respect to Feder and Vardi.
References
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Book
Computers and Intractability: A Guide to the Theory of NP-Completeness
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
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