It has been shown that for a general-valued constraint language Γ the following statements are equivalent: (1) any instance of VCSP(Γ) can be solved to optimality using a constant level of the Sherali-Adams LP hierarchy; (2) any instance of VCSP(Γ) can be solved to optimality using the third level of the Sherali-Adams LP hierarchy; (3) the support of Γ satisfies the “bounded width condition”, i.e., it contains weak near-unanimity operations of all arities.

We give a precise algebraic characterization of the power of Sherali--Adams relaxations for solvability of valued constraint satisfaction problems (CSPs) to optimality. The condition is that of bounded width, which has already been shown to capture the power of local consistency methods for decision CSPs and the power of semidefinite programming for robust approximation of CSPs. Our characterization has several algorithmic and complexity consequences. On the algorithmic side, we show that several novel and well-known valued constraint languages are tractable via the third level of the Sherali--Adams relaxation. For the known languages, this is a significantly simpler algorithm than those previously obtained. On the complexity side, we obtain a dichotomy theorem for valued constraint languages that can express an injective unary function. This implies a simple proof of the dichotomy theorem for conservative valued constraint languages established by Kolmogorov and Živný [ J. ACM, 60 (2013), 10], and also a dichotomy theorem for the exact solvability of minimum-solution problems. These are generalizations of minimum-ones problems to arbitrary finite domains. Our result improves on several previous classifications by Khanna et al. [ SIAM J. Comput., 30 (2001), pp. 1863--1920], Jonsson, Kuivinen, and Nordh [ SIAM J. Comput., 38 (2008), pp. 329--365], and Uppman [ Proceedings of ICALP'13, Springer, Berlin, 2013, pp. 804--815].

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).

Let $D$, called the domain, be a fixed finite set and let $\Gamma$, called the valued constraint language, be a fixed set of functions of the form $f:D^m\to\mathbb{Q}\cup\{\infty\}$, where different functions might have different arity $m$. We study the valued constraint satisfaction problem parametrized by $\Gamma$, denoted by VCSP$(\Gamma)$. These are minimization problems given by $n$ variables and the objective function given by a sum of functions from $\Gamma$, each depending on a subset of the $n$ variables. For example, if $D=\{0,1\}$ and $\Gamma$ contains all ternary $\{0,\infty\}$-valued functions, VCSP($\Gamma$) corresponds to 3-SAT. More generally, if $\Gamma$ contains only $\{0,\infty\}$-valued functions, VCSP($\Gamma$) corresponds to CSP($\Gamma$). If $D=\{0,1\}$ and $\Gamma$ contains all ternary $\{0,1\}$-valued functions, VCSP($\Gamma$) 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 $\Gamma$, BLP is a decision procedure for $\Gamma$ if and only if $\Gamma$ admits a symmetric fractional polymorphism of every arity. For a finite-valued constraint language $\Gamma$, BLP is a decision procedure if and only if $\Gamma$ admits a symmetric fractional polymorphism of some arity, or equivalently, if $\Gamma$ 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.

The connection between constraint languages and clone theory has been a fruitful line of research on the complexity of constraint satisfaction problems. In a recent result, Cohen et al. [SIAM J. Comput., 42 (2013), pp. 915--1939] have characterized a Galois connection between valued constraint languages and so-called weighted clones. In this paper, we study the structure of weighted clones. We extend the results of Creed and Živný from [Proceedings of the 17th International Conference on Principles and Practice of Constraint Programming, 2011, pp. 210--224] on types of weightings necessarily contained in every nontrivial weighted clone. This result has immediate computational complexity consequences as it provides necessary conditions for tractability of weighted clones and thus valued constraint languages. We demonstrate that some of the necessary conditions are also sufficient for tractability, while others are provably not.

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 languages defined on domains of arbitrary finite size.

A famous result by Jeavons, Cohen, and Gyssens shows that every Constraint Satisfaction Problem (CSP) where the constraints are preserved by a semi-lattice operation can be solved in polynomial time. This is one of the basic facts for the so-called universal algebraic approach to a systematic theory of tractability and hardness in finite domain constraint satisfaction. Not surprisingly, the theorem of Jeavons et al. fails for arbitrary infinite domain CSPs. Many CSPs of practical interest, though, and in particular those CSPs that are motivated by qualitative reasoning calculi from artificial intelligence, can be formulated with constraint languages that are rather well-behaved from a model-theoretic point of view. In particular, the automorphism group of these constraint languages tends to be large in the sense that the number of orbits of n -subsets of the automorphism group is bounded by some function in n . In this article we present a generalization of the theorem by Jeavons et al. to infinite domain CSPs where the number of orbits of n -subsets grows subexponentially in n , and prove that preservation under a semi-lattice operation for such CSPs implies polynomial-time tractability. Unlike the result of Jeavons et al., this includes CSPs that cannot be solved by Datalog.

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) sub modular on arbitrary lattices, (2) bisubmodular (also known as k-sub modular) on arbitrary finite domains, (3) weakly (and hence strongly) tree-sub modular on arbitrary trees.