--- _id: '644' abstract: - lang: eng text: 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 6= 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 f0;1g 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 finitevalued 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. author: - first_name: Vladimir full_name: Kolmogorov, Vladimir id: 3D50B0BA-F248-11E8-B48F-1D18A9856A87 last_name: Kolmogorov - first_name: Andrei full_name: Krokhin, Andrei last_name: Krokhin - first_name: Michal full_name: Rolinek, Michal id: 3CB3BC06-F248-11E8-B48F-1D18A9856A87 last_name: Rolinek citation: ama: Kolmogorov V, Krokhin A, Rolinek M. The complexity of general-valued CSPs. SIAM Journal on Computing. 2017;46(3):1087-1110. doi:10.1137/16M1091836 apa: Kolmogorov, V., Krokhin, A., & Rolinek, M. (2017). The complexity of general-valued CSPs. SIAM Journal on Computing. SIAM. https://doi.org/10.1137/16M1091836 chicago: Kolmogorov, Vladimir, Andrei Krokhin, and Michal Rolinek. “The Complexity of General-Valued CSPs.” SIAM Journal on Computing. SIAM, 2017. https://doi.org/10.1137/16M1091836. ieee: V. Kolmogorov, A. Krokhin, and M. Rolinek, “The complexity of general-valued CSPs,” SIAM Journal on Computing, vol. 46, no. 3. SIAM, pp. 1087–1110, 2017. ista: Kolmogorov V, Krokhin A, Rolinek M. 2017. The complexity of general-valued CSPs. SIAM Journal on Computing. 46(3), 1087–1110. mla: Kolmogorov, Vladimir, et al. “The Complexity of General-Valued CSPs.” SIAM Journal on Computing, vol. 46, no. 3, SIAM, 2017, pp. 1087–110, doi:10.1137/16M1091836. short: V. Kolmogorov, A. Krokhin, M. Rolinek, SIAM Journal on Computing 46 (2017) 1087–1110. date_created: 2018-12-11T11:47:40Z date_published: 2017-06-29T00:00:00Z date_updated: 2023-02-23T10:07:49Z day: '29' department: - _id: VlKo doi: 10.1137/16M1091836 ec_funded: 1 intvolume: ' 46' issue: '3' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1502.07327 month: '06' oa: 1 oa_version: Preprint page: 1087 - 1110 project: - _id: 25FBA906-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '616160' name: 'Discrete Optimization in Computer Vision: Theory and Practice' publication: SIAM Journal on Computing publication_status: published publisher: SIAM publist_id: '7138' quality_controlled: '1' related_material: record: - id: '1637' relation: other status: public scopus_import: 1 status: public title: The complexity of general-valued CSPs type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 46 year: '2017' ...