{"scopus_import":1,"publisher":"ACM","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2018-12-11T11:59:48Z","oa_version":"Preprint","oa":1,"status":"public","quality_controlled":"1","title":"The complexity of conservative valued CSPs","date_updated":"2021-01-12T07:00:00Z","month":"04","day":"02","type":"journal_article","abstract":[{"lang":"eng","text":"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."}],"main_file_link":[{"url":"http://arxiv.org/abs/1110.2809","open_access":"1"}],"department":[{"_id":"VlKo"}],"issue":"2","date_published":"2013-04-02T00:00:00Z","publication_status":"published","year":"2013","publist_id":"3971","volume":60,"intvolume":"        60","external_id":{"arxiv":["1110.2809"]},"author":[{"first_name":"Vladimir","last_name":"Kolmogorov","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","full_name":"Kolmogorov, Vladimir"},{"last_name":"Živný","first_name":"Stanislav","full_name":"Živný, Stanislav"}],"publication":"Journal of the ACM","doi":"10.1145/2450142.2450146","article_number":"10","_id":"2828","language":[{"iso":"eng"}],"citation":{"short":"V. Kolmogorov, S. Živný, Journal of the ACM 60 (2013).","ama":"Kolmogorov V, Živný S. The complexity of conservative valued CSPs. <i>Journal of the ACM</i>. 2013;60(2). doi:<a href=\"https://doi.org/10.1145/2450142.2450146\">10.1145/2450142.2450146</a>","apa":"Kolmogorov, V., &#38; Živný, S. (2013). The complexity of conservative valued CSPs. <i>Journal of the ACM</i>. ACM. <a href=\"https://doi.org/10.1145/2450142.2450146\">https://doi.org/10.1145/2450142.2450146</a>","chicago":"Kolmogorov, Vladimir, and Stanislav Živný. “The Complexity of Conservative Valued CSPs.” <i>Journal of the ACM</i>. ACM, 2013. <a href=\"https://doi.org/10.1145/2450142.2450146\">https://doi.org/10.1145/2450142.2450146</a>.","ista":"Kolmogorov V, Živný S. 2013. The complexity of conservative valued CSPs. Journal of the ACM. 60(2), 10.","ieee":"V. Kolmogorov and S. Živný, “The complexity of conservative valued CSPs,” <i>Journal of the ACM</i>, vol. 60, no. 2. ACM, 2013.","mla":"Kolmogorov, Vladimir, and Stanislav Živný. “The Complexity of Conservative Valued CSPs.” <i>Journal of the ACM</i>, vol. 60, no. 2, 10, ACM, 2013, doi:<a href=\"https://doi.org/10.1145/2450142.2450146\">10.1145/2450142.2450146</a>."}}