The Power of Linear Programming for Finite-Valued CSPs: A Constructive Characterization
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A class of valued constraint satisfaction problems (VCSPs) is characterised by a valued constraint language, a fixed set of cost functions on a finite domain. An instance of the problem is specified by a sum of cost functions from the language with the goal to minimise the sum.
We study which classes of finite-valued languages can be solved exactly by the basic linear programming relaxation (BLP). Thapper and Živný showed  that if BLP solves the language then the language admits a binary commutative fractional polymorphism. We prove that the converse is also true. This leads to a necessary and a sufficient condition which can be checked in polynomial time for a given language. In contrast, the previous necessary and sufficient condition due to  involved infinitely many inequalities.
More recently, Thapper and Živný  showed (using, in particular, a technique introduced in this paper) that core languages that do not satisfy our condition are NP-hard. Taken together, these results imply that a finite-valued language can either be solved using Linear Programming or is NP-hard.
KeywordsCost Function Constraint Satisfaction Problem Submodular Function Core Language Symmetric Operation
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