Abstract
We study the computational complexity of exact minimization of rational-valued discrete functions. Let Γ be a set of rational-valued functions on a fixed finite domain; such a set is called afinite-valued constraint language. The valued constraint satisfaction problem, VCSP(Γ), is the problem of minimizing a function given as a sum of functions from Γ. We establish a dichotomy theorem with respect to exact solvability forallfinite-valued constraint languages defined on domains ofarbitraryfinite size.We show that every constraint language Γ either admits a binary symmetric fractional polymorphism, in which case the basic linear programming relaxation solves any instance of VCSP(Γ) exactly, or Γ satisfies a simple hardness condition that allows for a polynomial-time reduction from Max-Cut to VCSP(Γ).
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