Generalising tractable VCSPs defined by symmetric tournament pair multimorphisms

We study optimisation problems that can be formulated as valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a \emph{constraint language}, a fixed set of cost functions taking finite and infinite costs 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 are interested in \emph{tractable} constraint languages; that is, languages that give rise to VCSP instances solvable in polynomial time. Cohen et al. (AIJ'06) have shown that constraint languages that admit the MJN multimorphism are tractable. Moreover, using a minimisation algorithm for submodular functions, Cohen et al. (TCS'08) have shown that constraint languages that admit an STP (symmetric tournament pair) multimorphism are tractable. We generalise these results by showing that languages admitting the MJN multimorphism on a subdomain and an STP multimorphisms on the complement of the subdomain are tractable. The algorithm is a reduction to the algorithm for languages admitting an STP multimorphism.Comment: 14 page

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

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

Complexity Classifications for the Valued Constraint Satisfaction Problem

In a valued constraint satisfaction problem (VCSP), the goal is to find an assignment of values to variables that minimizes a given sum of functions. Each function in the sum depends on a subset of variables, takes values which are rational numbers or infinity, and is chosen from a fixed finite set of functions called a constraint language. We study how the computational complexity of this problem depends on the constraint language. We often consider the case where infinite values are disallowed, and refer to such constraint languages as being finite-valued. If we consider such finite-valued constraint languages, the case where we allow variables to take two values was classified by Cohen et al., who show that submodular functions essentially give rise to the only tractable case. Non-submodular functions can be used to express the NP-hard Max Cut problem. We consider the case where the variables can take three values, and identify a new infinite set of functions called skew bisubmodular functions which imply tractability. We prove that submodularity with respect to some total order and skew bisubmodularity give rise to the only tractable cases, and in all other cases Max Cut can be expressed. We also show that our characterisation of tractable cases is tight, that is, none of the conditions can be omitted. Thus, our results provide a new dichotomy theorem in constraint satisfaction research. We also negatively answer the question of whether multimorphisms can capture all necessary tractable constraint languages. We then study the VCSP as a homomorphism problem on digraphs. By adapting a proof designed for CSPs we show that each VCSP with a fixed finite constraint language is equivalent to one where the constraint language consists of one {0,infinity}-valued binary function (i.e. a digraph) and one finite-valued unary function. This latter problem is known as the Minimum Cost Homomorphism Problem for digraphs. We also show that our reduction preserves a number of useful algebraic properties of the constraint language. Finally, given a finite-valued constraint language, we consider the case where the variables of our VCSP are allowed to take four values. We prove that 1-defect chain multimorphisms, which are required in the four element dichotomy of Min CSP, are a special case of more general fractional polymorphisms we call {a,b}-1-defect fractional polymorphisms. We conclude with a conjecture for the four element case, and some interesting open problems which might lead to a tighter description of tractable finite-valued constraint languages on finite domains of any size

An algebraic theory of complexity for valued constraints: Establishing a Galois connection

Abstract. The complexity of any optimisation problem depends critically on the form of the objective function. Valued constraint satisfaction problems are discrete optimisation problems where the function to be minimised is given as a sum of cost functions defined on specified subsets of variables. These cost functions are chosen from some fixed set of available cost functions, known as a valued constraint language. We show in this paper that when the costs are non-negative rational numbers or infinite, then the complexity of a valued constraint problem is determined by certain algebraic properties of this valued constraint language, which we call weighted polymorphisms. We define a Galois connection between valued constraint languages and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterised. These results provide a new approach in the search for tractable valued constraint languages

The complexity of approximating conservative counting CSPs

We study the complexity of approximately solving the weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, there is a classification known for the case in which the domain of functions in F is Boolean. In this paper, we give a classification for the more general problem where functions in F have an arbitrary finite domain. We define the notions of weak log-modularity and weak log-supermodularity. We show that if F is weakly log-modular, then #CSP(F)is in FP. Otherwise, it is at least as difficult to approximate as #BIS, the problem of counting independent sets in bipartite graphs. #BIS is complete with respect to approximation-preserving reductions for a logically-defined complexity class #RHPi1, and is believed to be intractable. We further sub-divide the #BIS-hard case. If F is weakly log-supermodular, then we show that #CSP(F) is as easy as a (Boolean) log-supermodular weighted #CSP. Otherwise, we show that it is NP-hard to approximate. Finally, we give a full trichotomy for the arity-2 case, where #CSP(F) is in FP, or is #BIS-equivalent, or is equivalent in difficulty to #SAT, the problem of approximately counting the satisfying assignments of a Boolean formula in conjunctive normal form. We also discuss the algorithmic aspects of our classification.Comment: Minor revisio