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. This framework includes and generalises well-studied constraint satisfaction problems (CSPs) and maximum constraint satisfaction problems (Max-CSPs). Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation. Using this result, we obtain tractability of several novel and previously widely-open classes of VCSPs, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) bisubmodular (also known as k-submodular) on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: Corrected a few typo

We study the computational complexity of exact minimisation of rational-valued discrete functions. Let Γ be a set of rational-valued functions on a fixed finite domain; such a set is called a finite-valued constraint language. The valued constraint satisfaction problem, VCSP(Γ), is the problem of minimising a function given as a sum of functions from Γ. We establish a dichotomy theorem with respect to exact solvability for all finite-valued constraint languages defined on domains of arbitrary finite 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).

Let D, called the domain, be a fixed finite set and let Γ, called the valued constraint language, be a fixed set of functions of the form f : D m → Q ∪ {∞}, where different functions might have different arity m. We study the valued constraint satisfaction problem parametrized by Γ, denoted by VCSP(Γ). These are minimization problems given by n variables and the objective function given by a sum of functions from Γ, each depending on a subset of the n variables. For example, if D = {0, 1} and Γ contains all ternary {0, ∞}-valued functions, VCSP(Γ) corresponds to 3-SAT. More generally, if Γ contains only {0, ∞}-valued functions, VCSP(Γ) corresponds to CSP(Γ). If D = {0, 1} and Γ contains all ternary {0, 1}-valued functions, VCSP(Γ) corresponds to Min-3-SAT, in which the goal is to minimize the number of unsatisfied clauses in a 3-CNF instance. Finite-valued constraint languages contain functions that take on only rational values and not infinite values.Our main result is a precise algebraic characterization of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation (BLP). For a valued constraint language Γ, BLP is a decision procedure for Γ if and only if Γ admits a symmetric fractional polymorphism of every arity. For a finite-valued constraint language Γ, BLP is a decision procedure if and only if Γ admits a symmetric fractional polymorphism of some arity, or equivalently, if Γ admits a symmetric fractional polymorphism of arity 2. Using these results, we obtain tractability of several novel classes of problems, including problems over valued constraint languages that are (1) submodular on arbitrary lattices; (2) k-submodular on arbitrary finite domains; (3) weakly (and hence strongly) tree submodular on arbitrary trees.

We study the computational complexity of exact minimisation of rational-valued discrete functions. Let Γ be a set of rational-valued functions on a fixed finite domain; such a set is called a finite-valued constraint language. The valued constraint satisfaction problem, VCSP(Γ), is the problem of minimising a function given as a sum of functions from Γ. We establish a dichotomy theorem with respect to exact solvability for all finite-valued constraint languages defined on domains of arbitrary finite 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).