# The quadratic M-convexity testing problem

## TL;DR

This paper investigates the quadratic M-convexity testing problem, establishing its computational complexity and providing an efficient algorithm for cases where it is polynomial-time solvable, thus advancing understanding in discrete convex analysis.

## Contribution

The paper proves that QMCTP is co-NP-complete generally but polynomial-time solvable under a natural assumption, and introduces an $O(n^2)$ algorithm for such cases.

## Key findings

- QMCTP is co-NP-complete in general.
- QMCTP is polynomial-time solvable under a natural assumption.
- An $O(n^2)$-time algorithm is proposed for the polynomial-time solvable case.

## Abstract

M-convex functions, which are a generalization of valuated matroids, play a central role in discrete convex analysis. Quadratic M-convex functions constitute a basic and important subclass of M-convex functions, which has a close relationship with phylogenetics as well as valued constraint satisfaction problems. In this paper, we consider the quadratic M-convexity testing problem (QMCTP), which is the problem of deciding whether a given quadratic function on $\{0,1\}^n$ is M-convex. We show that QMCTP is co-NP-complete in general, but is polynomial-time solvable under a natural assumption. Furthermore, we propose an $O(n^2)$-time algorithm for solving QMCTP in the polynomial-time solvable case.