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).

We study the complexity of valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a constraint language, a fixed set of cost functions over a finite domain. An instance of the problem is specified by a sum of cost functions from the language and the goal is to minimise the sum. Under the unique games conjecture, the approximability of finite-valued VCSPs is wellunderstood, see Raghavendra [FOCS'08]. However, there is no characterisation of finite-valued VCSPs, let alone general-valued VCSPs, that can be solved exactly in polynomial time, thus giving insights from a combinatorial optimisation perspective.We consider the case of languages containing all possible unary cost functions. We prove a Schaefer-like dichotomy theorem for conservative valued languages: if all cost functions in the language satisfy a certain condition (specified by a complementary combination of STP and MJN multimorphisms), then any instance can be solved in polynomial time (via a new algorithm developed in this paper), otherwise the language is NP-hard. This is the first complete complexity classification of general-valued constraint languages over non-Boolean domains. It is a common phenomenon that complexity classifications of problems over non-Boolean domains is significantly harder than the Boolean case. The polynomial-time algorithm we present for the tractable cases is a generalisation of the submodular minimisation problem and a result of Cohen et al. [TCS'08].Our results generalise previous results by Takhanov [STACS'10] and (a subset of results) by Cohen et al. [AIJ'06] and Deineko et al. [JACM'08]. Moreover, our results do not rely on any computerassisted search as in Deineko et al. [JACM'08], and provide a powerful tool for proving hardness of finite-valued and general-valued languages.

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).