### The complexity of conservative valued CSPsThe complexity of conservative valued CSPs

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 Author Kolmogorov, Vladimir ♦ Ivn, Stanislav Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2013 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Complexity ♦ Dichotomy ♦ Discrete optimization ♦ Multimorphisms ♦ Submodularity ♦ Valued constraint satisfaction problems 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 $\textit{conservative}$ and studied by Bulatov [2003, 2011] and recently by Barto [2011]. Since we study valued languages, we call a language $\textit{conservative}$ 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 multimorphisms), 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. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2013-05-03 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 60 Issue Number 2 Page Count 38 Starting Page 1 Ending Page 38

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Source: ACM Digital Library