Submodular Functions and Valued Constraint Satisfaction Problems over Infinite Domains

TL;DR

This paper explores the complexity of infinite-domain VCSPs with piecewise linear homogeneous cost functions, establishing polynomial-time solvability for submodular cases and NP-hardness when non-submodular functions are introduced.

Contribution

It introduces a systematic study of infinite-domain VCSPs, identifying submodularity as the key property for tractability in this setting.

Findings

Submodular cost functions lead to polynomial-time solvable VCSPs.

Adding non-submodular functions makes VCSPs NP-hard.

This work extends VCSP complexity classification to infinite domains.

Abstract

Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. It is desirable to classify the computational complexity of VCSPs depending on a fixed set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified in this sense. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of infinite-domain VCSPs by studying the complexity of VCSPs for piecewise linear homogeneous cost functions. We show that such VCSPs can be solved in polynomial time when the cost functions are additionally submodular, and that this is indeed a maximally tractable class: adding any cost function that is not submodular leads to an NP-hard VCSP.