Piecewise Linear Valued CSPs Solvable by Linear Programming Relaxation

TL;DR

This paper investigates the complexity of infinite-domain VCSPs with piecewise linear homogeneous cost functions, showing they are solvable via linear programming relaxation, especially for submodular functions.

Contribution

It introduces a systematic study of infinite-domain VCSPs with PLH cost functions and establishes polynomial-time solvability using linear programming relaxation techniques.

Findings

VCSPs with submodular PLH cost functions are solvable in polynomial time.

Reduction of infinite-domain VCSPs to finite-domain VCSPs solvable by linear programming.

Submodular PLH functions form a maximally tractable class.

Abstract

Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. The computational complexity of VCSPs depends on the 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. 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. Such VCSPs can be solved in polynomial time if the cost functions are improved by fully symmetric fractional operations of all arities. We show this by reducing the problem to a finite-domain VCSP which can be solved using the basic linear program relaxation. It follows that VCSPs for submodular PLH cost functions can be solved in polynomial time; in fact, we show that submodular PLH functions form a maximally tractable class of PLH cost functions.