The Power of Linear Programming for General-Valued CSPs

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 → Q ∪ {∞}, where different functions might have different arity m. We study the valued constraint satisfaction problem parametrised by Γ, denoted by VCSP(Γ). These are minimisation 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 minimise 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 characterisation 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.