Complexity Classifications for the Valued Constraint Satisfaction Problem

Abstract

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 -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 -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.