{"main_file_link":[{"url":"http://arxiv.org/abs/1008.1555","open_access":"1"}],"_id":"3284","publication_status":"published","day":"01","extern":1,"conference":{"name":"SODA: Symposium on Discrete Algorithms"},"status":"public","date_updated":"2021-01-12T07:42:23Z","date_published":"2012-01-01T00:00:00Z","citation":{"ama":"Kolmogorov V, Živný S. The complexity of conservative valued CSPs. In: SIAM; 2012:750-759.","ieee":"V. Kolmogorov and S. Živný, “The complexity of conservative valued CSPs,” presented at the SODA: Symposium on Discrete Algorithms, 2012, pp. 750–759.","mla":"Kolmogorov, Vladimir, and Stanislav Živný. <i>The Complexity of Conservative Valued CSPs</i>. SIAM, 2012, pp. 750–59.","chicago":"Kolmogorov, Vladimir, and Stanislav Živný. “The Complexity of Conservative Valued CSPs,” 750–59. SIAM, 2012.","apa":"Kolmogorov, V., &#38; Živný, S. (2012). The complexity of conservative valued CSPs (pp. 750–759). Presented at the SODA: Symposium on Discrete Algorithms, SIAM.","ista":"Kolmogorov V, Živný S. 2012. The complexity of conservative valued CSPs. SODA: Symposium on Discrete Algorithms, 750–759.","short":"V. Kolmogorov, S. Živný, in:, SIAM, 2012, pp. 750–759."},"quality_controlled":0,"title":"The complexity of conservative valued CSPs","date_created":"2018-12-11T12:02:27Z","publist_id":"3362","month":"01","publisher":"SIAM","type":"conference","oa":1,"acknowledgement":"Vladimir Kolmogorov is supported by the Royal Academy of Eng ineering/EPSRC.","page":"750 - 759","abstract":[{"lang":"eng","text":"We study the complexity of valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised 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 minimise the sum. Under the unique games conjecture, the approximability of finite-valued VCSPs is well-understood, see Raghavendra [FOCS’08]. However, there is no characterisation of finite-valued VCSPs, let alone general-valued VCSPs, that can be solved exactly in polynomial time, thus giving insights from a combinatorial optimisation perspective.\nWe 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 [LICS’03] and recently by Barto [LICS’11]. 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. [AIJ’06] for languages over Boolean domains, by Deineko et al. [JACM’08] for {0,1}-valued languages (a.k.a Max-CSP), and by Takhanov [STACS’10] for {0,∞}-valued languages containing all finite- valued unary cost functions (a.k.a. Min-Cost-Hom).\nWe 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 paper), 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 is significantly harder than the Boolean case. The polynomial-time algorithm we present for the tractable cases is a generalisation of the submodular minimisation problem and a result of Cohen et al. [TCS’08].\nOur results generalise previous results by Takhanov [STACS’10] and (a subset of results) by Cohen et al. [AIJ’06] and Deineko et al. [JACM’08]. Moreover, our results do not rely on any computer-assisted search as in Deineko et al. [JACM’08], and provide a powerful tool for proving hardness of finite-valued and general-valued languages."}],"author":[{"last_name":"Kolmogorov","first_name":"Vladimir","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","full_name":"Vladimir Kolmogorov"},{"first_name":"Stanislav","last_name":"Živný","full_name":"Živný, Stanislav"}],"year":"2012"}