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The complexity of general-valued CSPs

Kolmogorov, V.; Krokhin, A.; Rolínek, M.

The complexity of general-valued CSPs Thumbnail


V. Kolmogorov

M. Rolínek


An instance of the valued constraint satisfaction problem (VCSP) is given by a finite set of variables, a finite domain of labels, and a sum of functions, each function depending on a subset of the variables. Each function can take finite values specifying costs of assignments of labels to its variables or the infinite value, which indicates an infeasible assignment. The goal is to find an assignment of labels to the variables that minimizes the sum. We study, assuming that P $\ne$ NP, how the complexity of this very general problem depends on the set of functions allowed in the instances, the so-called constraint language. The case when all allowed functions take values in $\{0,\infty\}$ corresponds to ordinary CSPs, where one deals only with the feasibility issue, and there is no optimization. This case is the subject of the algebraic CSP dichotomy conjecture predicting for which constraint languages CSPs are tractable (i.e., solvable in polynomial time) and for which they are NP-hard. The case when all allowed functions take only finite values corresponds to a finite-valued CSP, where the feasibility aspect is trivial and one deals only with the optimization issue. The complexity of finite-valued CSPs was fully classified by Thapper and Živný. An algebraic necessary condition for tractability of a general-valued CSP with a fixed constraint language was recently given by Kozik and Ochremiak. As our main result, we prove that if a constraint language satisfies this algebraic necessary condition, and the feasibility CSP (i.e., the problem of deciding whether a given instance has a feasible solution) corresponding to the VCSP with this language is tractable, then the VCSP is tractable. The algorithm is a simple combination of the assumed algorithm for the feasibility CSP and the standard LP relaxation. As a corollary, we obtain that a dichotomy for ordinary CSPs would imply a dichotomy for general-valued CSPs.


Kolmogorov, V., Krokhin, A., & Rolínek, M. (2017). The complexity of general-valued CSPs. SIAM Journal on Computing, 46(3), 1087-1110.

Journal Article Type Article
Acceptance Date Feb 8, 2017
Online Publication Date Jun 29, 2017
Publication Date Jul 1, 2017
Deposit Date Feb 20, 2017
Publicly Available Date Jun 29, 2017
Journal SIAM Journal on Computing
Print ISSN 0097-5397
Electronic ISSN 1095-7111
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 46
Issue 3
Pages 1087-1110
Related Public URLs


Accepted Journal Article (382 Kb)

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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

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