The power of linear programming for finite-valued CSPs: A constructive characterization LNCS Kolmogorov, Vladimir 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 [20] 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 [20] involved infinitely many inequalities. More recently, Thapper and Živný [21] 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. Springer 2013 info:eu-repo/semantics/conferenceObject doc-type:conferenceObject text http://purl.org/coar/resource_type/c_5794 https://research-explorer.ista.ac.at/record/2518 Kolmogorov V. The power of linear programming for finite-valued CSPs: A constructive characterization. In: Vol 7965. Springer; 2013:625-636. doi:<a href="https://doi.org/10.1007/978-3-642-39206-1_53">10.1007/978-3-642-39206-1_53</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1007/978-3-642-39206-1_53 info:eu-repo/semantics/altIdentifier/arxiv/1207.7213 info:eu-repo/semantics/openAccess