In:
Journal of the ACM, Association for Computing Machinery (ACM), Vol. 60, No. 2 ( 2013-04), p. 1-38
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 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.
Type of Medium:
Online Resource
ISSN:
0004-5411
,
1557-735X
DOI:
10.1145/2450142.2450146
Language:
English
Publisher:
Association for Computing Machinery (ACM)
Publication Date:
2013
detail.hit.zdb_id:
2006500-0
detail.hit.zdb_id:
6759-3
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