A tractable class of binary VCSPs via M-convex intersection

TL;DR

This paper identifies a larger class of binary VCSPs that can be solved efficiently by representing them as the sum of two quadratic M-convex functions, extending previous tractability results.

Contribution

It provides a polynomial-time algorithm to represent certain binary VCSPs as sums of two quadratic M-convex functions, expanding the known tractable classes.

Findings

Larger tractable class of binary VCSPs identified

Algorithm for concrete representation of VCSPs as M-convex sums

Extension beyond the joint winner property (JWP) class

Abstract

A binary VCSP is a general framework for the minimization problem of a function represented as the sum of unary and binary cost functions. An important line of VCSP research is to investigate what functions can be solved in polynomial time. Cooper and \v{Z}ivn\'{y} classified the tractability of binary VCSP instances according to the concept of "triangle," and showed that the only interesting tractable case is the one induced by the joint winner property (JWP). Recently, Iwamasa, Murota, and \v{Z}ivn\'{y} made a link between VCSP and discrete convex analysis, showing that a function satisfying the JWP can be transformed into a function represented as the sum of two quadratic M-convex functions, which can be minimized in polynomial time via an M-convex intersection algorithm if the value oracle of each M-convex function is given. In this paper, we give an algorithmic answer to a natural question: What binary finite-valued CSP instances can be represented as the sum of two quadratic M-convex functions and can be solved in polynomial time via an M-convex intersection algorithm? We solve this problem by devising a polynomial-time algorithm for obtaining a concrete form of the representation in the representable case. Our result presents a larger tractable class of binary finite-valued CSPs, which properly contains the JWP class.