A dichotomy theorem for conservative general-valued CSPs

TL;DR

This paper establishes a clear complexity classification for conservative valued constraint satisfaction problems, showing that they are either solvable in polynomial time or NP-hard based on specific algebraic conditions.

Contribution

It proves a dichotomy theorem for conservative VCSPs, extending previous results by characterizing tractability using multimorphisms and unifying earlier classifications.

Findings

Polynomial-time solvability when all cost functions satisfy the specified condition.

NP-hardness for languages not satisfying the condition.

Generalizes previous results on finite-valued and $ ext{0,} ext{ extonehalf} ext{ extminus} ext{ extonehalf}$-valued languages.

Abstract

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. We prove a Schaefer-like dichotomy theorem for this case: if all cost functions in the language satisfy a certain condition (specified by a complementary combination of \emph{STP and MJN multimorphisms}) then any instance can be solved in polynomial time by the algorithm of Kolmogorov and Zivny (arXiv:1008.3104v1), otherwise the language is NP-hard. This generalises recent results of Takhanov (STACS'10) who considered $\{0,\infty\}$-valued languages containing additionally all finite-valued unary cost functions, and Kolmogorov and Zivny (arXiv:1008.1555v1) who considered \emph{finite-valued} conservative languages.