The complexity of conservative valued CSPs

V. Kolmogorov, S. Živný, Journal of the ACM 60 (2013).


Journal Article | Published | English
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Abstract
We study the complexity of valued constraint satisfaction problems (VCSPs) parametrized by a constraint language, a fixed set of cost functions over a finite domain. An instance of the problem is specified by a sum of cost functions from the language and the goal is to minimize the sum. Under the unique games conjecture, the approximability of finite-valued VCSPs is well understood, see Raghavendra [2008]. However, there is no characterization of finite-valued VCSPs, let alone general-valued VCSPs, that can be solved exactly in polynomial time, thus giving insights from a combinatorial optimization perspective. We consider the case of languages containing all possible unary cost functions. In the case of languages consisting of only {0, ∞}-valued cost functions (i.e., relations), such languages have been called conservative and studied by Bulatov [2003, 2011] and recently by Barto [2011]. Since we study valued languages, we call a language conservative if it contains all finite-valued unary cost functions. The computational complexity of conservative valued languages has been studied by Cohen et al. [2006] for languages over Boolean domains, by Deineko et al. [2008] for {0, 1}-valued languages (a.k.a Max-CSP), and by Takhanov [2010a] for {0, ∞}-valued languages containing all finite-valued unary cost functions (a.k.a. Min-Cost-Hom). We prove a Schaefer-like dichotomy theorem for conservative valued languages: if all cost functions in the language satisfy a certain condition (specified by a complementary combination of STP and MJN multimor-phisms), then any instance can be solved in polynomial time (via a new algorithm developed in this article), otherwise the language is NP-hard. This is the first complete complexity classification of general-valued constraint languages over non-Boolean domains. It is a common phenomenon that complexity classifications of problems over non-Boolean domains are significantly harder than the Boolean cases. The polynomial-time algorithm we present for the tractable cases is a generalization of the submodular minimization problem and a result of Cohen et al. [2008]. Our results generalize previous results by Takhanov [2010a] and (a subset of results) by Cohen et al. [2006] and Deineko et al. [2008]. Moreover, our results do not rely on any computer-assisted search as in Deineko et al. [2008], and provide a powerful tool for proving hardness of finite-valued and general-valued languages.
Publishing Year
Date Published
2013-04-02
Journal Title
Journal of the ACM
Volume
60
Issue
2
Article Number
10
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Cite this

Kolmogorov V, Živný S. The complexity of conservative valued CSPs. Journal of the ACM. 2013;60(2). doi:10.1145/2450142.2450146
Kolmogorov, V., & Živný, S. (2013). The complexity of conservative valued CSPs. Journal of the ACM, 60(2). https://doi.org/10.1145/2450142.2450146
Kolmogorov, Vladimir, and Stanislav Živný. “The Complexity of Conservative Valued CSPs.” Journal of the ACM 60, no. 2 (2013). https://doi.org/10.1145/2450142.2450146.
V. Kolmogorov and S. Živný, “The complexity of conservative valued CSPs,” Journal of the ACM, vol. 60, no. 2, 2013.
Kolmogorov V, Živný S. 2013. The complexity of conservative valued CSPs. Journal of the ACM. 60(2).
Kolmogorov, Vladimir, and Stanislav Živný. “The Complexity of Conservative Valued CSPs.” Journal of the ACM, vol. 60, no. 2, 10, ACM, 2013, doi:10.1145/2450142.2450146.

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