Weighted Triangle-free 2-matching Problem with Edge-disjoint Forbidden Triangles

TL;DR

This paper presents the first polynomial-time algorithm for the weighted triangle-free 2-matching problem when the forbidden triangles are edge-disjoint, using an extended formulation and polynomial-time separation.

Contribution

It introduces a novel polynomial-time algorithm for a long-standing open problem by developing an extended formulation with a polynomial-time separation procedure.

Findings

First polynomial-time algorithm for the problem

Extended formulation with exponentially many inequalities

Separation problem solvable in polynomial time

Abstract

The weighted $\mathcal{T}$-free $2$-matching problem is the following problem: given an undirected graph $G$, a weight function on its edge set, and a set $\mathcal{T}$ of triangles in $G$, find a maximum weight $2$-matching containing no triangle in $\mathcal{T}$. When $\mathcal{T}$ is the set of all triangles in $G$, this problem is known as the weighted triangle-free $2$-matching problem, which is a long-standing open problem. A main contribution of this paper is to give a first polynomial-time algorithm for the weighted $\mathcal{T}$-free $2$-matching problem under the assumption that $\mathcal{T}$ is a set of edge-disjoint triangles. In our algorithm, a key ingredient is to give an extended formulation representing the solution set, that is, we introduce new variables and represent the convex hull of the feasible solutions as a projection of another polytope in a higher dimensional space. Although our extended formulation has exponentially many inequalities, we show that the separation problem can be solved in polynomial time, which leads to a polynomial-time algorithm for the weighted $\mathcal{T}$-free $2$-matching problem.