# Discrete Convexity in Joint Winner Property

## TL;DR

This paper establishes a connection between joint winner property in valued constraint satisfaction problems and M-convexity in discrete convex analysis, introducing new algorithms for efficient minimization of Z-free functions.

## Contribution

It reveals the relationship between JWP and M-convexity, introduces the M-convex completion problem, and proposes a faster minimization algorithm for Z-free functions.

## Key findings

- Z-free functions can be minimized in polynomial time using M-convex intersection algorithms.
- A new algorithm for Z-free function minimization is faster for certain parameters.
- The paper bridges VCSPs and discrete convex analysis through the concept of M-convexity.

## Abstract

In this paper, we reveal a relation between joint winner property (JWP) in the field of valued constraint satisfaction problems (VCSPs) and M${}^\natural$-convexity in the field of discrete convex analysis (DCA). We introduce the M${}^\natural$-convex completion problem, and show that a function $f$ satisfying the JWP is Z-free if and only if a certain function $\overline{f}$ associated with $f$ is M${}^\natural$-convex completable. This means that if a function is Z-free, then the function can be minimized in polynomial time via M${}^\natural$-convex intersection algorithms. Furthermore we propose a new algorithm for Z-free function minimization, which is faster than previous algorithms for some parameter values.