@inproceedings{2518,
  abstract     = {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.},
  author       = {Kolmogorov, Vladimir},
  location     = {Riga, Latvia},
  number       = {1},
  pages        = {625 -- 636},
  publisher    = {Springer},
  title        = {{The power of linear programming for finite-valued CSPs: A constructive characterization}},
  doi          = {10.1007/978-3-642-39206-1_53},
  volume       = {7965},
  year         = {2013},
}