The complexity of finite-valued CSPs

We study the computational complexity of exact minimisation of rational-valued discrete functions. Let $\Gamma$ 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, $\operatorname{VCSP}(\Gamma)$, is the problem of minimising a function given as a sum of functions from $\Gamma$. 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 $\Gamma$ either admits a binary symmetric fractional polymorphism in which case the basic linear programming relaxation solves any instance of $\operatorname{VCSP}(\Gamma)$ exactly, or $\Gamma$ satisfies a simple hardness condition that allows for a polynomial-time reduction from Max-Cut to $\operatorname{VCSP}(\Gamma)$

The complexity of conservative finite-valued CSPs

We study the complexity of valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a \emph{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. We consider the case of so-called \emph{conservative} languages; that is, languages containing all unary cost functions, thus allowing arbitrary restrictions on the domains of the variables. This problem has been studied by Bulatov [LICS'03] for $\{0,\infty\}$-valued languages (i.e. CSP), by Cohen~\etal\ (AIJ'06) for Boolean domains, by Deineko et al. (JACM'08) for $\{0,1\}$-valued cost functions (i.e. Max-CSP), and by Takhanov (STACS'10) for $\{0,\infty\}$-valued languages containing all finite-valued unary cost functions (i.e. Min-Cost-Hom). We give an elementary proof of a complete complexity classification of conservative finite-valued languages: we show that every conservative finite-valued language is either tractable or NP-hard. This is the \emph{first} dichotomy result for finite-valued VCSPs over non-Boolean domains.Comment: 15 page

A Galois Connection for Weighted (Relational) Clones of Infinite Size

A Galois connection between clones and relational clones on a fixed finite domain is one of the cornerstones of the so-called algebraic approach to the computational complexity of non-uniform Constraint Satisfaction Problems (CSPs). Cohen et al. established a Galois connection between finitely-generated weighted clones and finitely-generated weighted relational clones [SICOMP'13], and asked whether this connection holds in general. We answer this question in the affirmative for weighted (relational) clones with real weights and show that the complexity of the corresponding valued CSPs is preserved

The Complexity of General-Valued CSPs

An instance of the Valued Constraint Satisfaction Problem (VCSP) is given by a finite set of variables, a finite domain of labels, and a sum of functions, each function depending on a subset of the variables. Each function can take finite values specifying costs of assignments of labels to its variables or the infinite value, which indicates an infeasible assignment. The goal is to find an assignment of labels to the variables that minimizes the sum. We study, assuming that P ≠ NP, how the complexity of this very general problem depends on the set of functions allowed in the instances, the so-called constraint language. The case when all allowed functions take values in {0, ∞} corresponds to ordinary CSPs, where one deals only with the feasibility issue and there is no optimization. This case is the subject of the Algebraic CSP Dichotomy Conjecture predicting for which constraint languages CSPs are tractable (i.e. solvable in polynomial time) and for which NP-hard. The case when all allowed functions take only finite values corresponds to finite-valued CSP, where the feasibility aspect is trivial and one deals only with the optimization issue. The complexity of finite-valued CSPs was fully classified by Thapper and Zivny. An algebraic necessary condition for tractability of a general-valued CSP with a fixed constraint language was recently given by Kozik and Ochremiak. As our main result, we prove that if a constraint language satisfies this algebraic necessary condition, and the feasibility CSP (i.e. the problem of deciding whether a given instance has a feasible solution) corresponding to the VCSP with this language is tractable, then the VCSP is tractable. The algorithm is a simple combination of the assumed algorithm for the feasibility CSP and the standard LP relaxation. As a corollary, we obtain that a dichotomy for ordinary CSPs would imply a dichotomy for general-valued CSPs

The power of Sherali-Adams relaxations for general-valued CSPs

We give a precise algebraic characterisation of the power of Sherali-Adams relaxations for solvability of valued constraint satisfaction problems 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 characterisation has several algorithmic and complexity consequences. On the algorithmic side, we show that several novel and many 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 the previously obtained ones. 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 Zivny [JACM'13], and also a dichotomy theorem for the exact solvability of Minimum-Solution problems. These are generalisations of Minimum-Ones problems to arbitrary finite domains. Our result improves on several previous classifications by Khanna et al. [SICOMP'00], Jonsson et al. [SICOMP'08], and Uppman [ICALP'13].Comment: Full version of an ICALP'15 paper (arXiv:1502.05301

Computational Complexity of the Minimum Cost Homomorphism Problem on Three-Element Domains

In this paper we study the computational complexity of the (extended) minimum cost homomorphism problem (Min-Cost-Hom) as a function of a constraint language, i.e. a set of constraint relations and cost functions that are allowed to appear in instances. A wide range of natural combinatorial optimisation problems can be expressed as Min-Cost-Homs and a classification of their complexity would be highly desirable, both from a direct, applied point of view as well as from a theoretical perspective. Min-Cost-Hom can be understood either as a flexible optimisation version of the constraint satisfaction problem (CSP) or a restriction of the (general-valued) valued constraint satisfaction problem (VCSP). Other optimisation versions of CSPs such as the minimum solution problem (Min-Sol) and the minimum ones problem (Min-Ones) are special cases of Min-Cost-Hom. The study of VCSPs has recently seen remarkable progress. A complete classification for the complexity of finite-valued languages on arbitrary finite domains has been obtained Thapper and Zivny [STOC'13]. However, understanding the complexity of languages that are not finite-valued appears to be more difficult. Min-Cost-Hom allows us to study problematic languages of this type without having to deal with with the full generality of the VCSP. A recent classification for the complexity of three-element Min-Sol, Uppman [ICALP'13], takes a step in this direction. In this paper we extend this result considerably by determining the complexity of three-element Min-Cost-Hom

Hybrid VCSPs with crisp and conservative valued templates

A constraint satisfaction problem (CSP) is a problem of computing a homomorphism ${\bf R} \rightarrow {\bf \Gamma}$ between two relational structures. Analyzing its complexity has been a very fruitful research direction, especially for fixed template CSPs, denoted $CSP({\bf \Gamma})$, in which the right side structure ${\bf \Gamma}$ is fixed and the left side structure ${\bf R}$ is unconstrained. Recently, the hybrid setting, written $CSP_{\mathcal{H}}({\bf \Gamma})$, where both sides are restricted simultaneously, attracted some attention. It assumes that ${\bf R}$ is taken from a class of relational structures $\mathcal{H}$ that additionally is closed under inverse homomorphisms. The last property allows to exploit algebraic tools that have been developed for fixed template CSPs. The key concept that connects hybrid CSPs with fixed-template CSPs is the so called "lifted language". Namely, this is a constraint language ${\bf \Gamma}_{{\bf R}}$ that can be constructed from an input ${\bf R}$. The tractability of that language for any input ${\bf R}\in\mathcal{H}$ is a necessary condition for the tractability of the hybrid problem. In the first part we investigate templates ${\bf \Gamma}$ for which the latter condition is not only necessary, but also is sufficient. We call such templates ${\bf \Gamma}$ widely tractable. For this purpose, we construct from ${\bf \Gamma}$ a new finite relational structure ${\bf \Gamma}'$ and define $\mathcal{H}_0$ as a class of structures homomorphic to ${\bf \Gamma}'$. We prove that wide tractability is equivalent to the tractability of $CSP_{\mathcal{H}_0}({\bf \Gamma})$. Our proof is based on the key observation that ${\bf R}$ is homomorphic to ${\bf \Gamma}'$ if and only if the core of ${\bf \Gamma}_{{\bf R}}$ is preserved by a Siggers polymorphism. Analogous result is shown for valued conservative CSPs.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1504.0706

The Power of Linear Programming for Valued CSPs

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

Necessary conditions for tractability of valued CSPs

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. [SICOMP'13] have characterised 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 Zivny from [CP'11/SICOMP'13] 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.Comment: To appear in SIAM Journal on Discrete Mathematics (SIDMA