Validating a PTAS for Triangle-Free 2-Matching via a Simple Decomposition Theorem

TL;DR

This paper introduces a simple decomposition theorem for triangle-free 2-matchings, simplifying the proof of a PTAS for finding maximum such matchings, which was previously complex to validate.

Contribution

The paper presents a natural and straightforward decomposition theorem that simplifies the proof of the PTAS for maximum triangle-free 2-matchings.

Findings

A simple decomposition theorem for triangle-free 2-matchings.

A more accessible proof of the PTAS's validity.

Enhanced understanding of the structure of triangle-free 2-matchings.

Abstract

A triangle-free (simple) 2-matching is an edge set that has at most $2$ edges incident to each vertex and contains no cycle of length $3$. For the problem of finding a maximum cardinality triangle-free 2-matching in a given graph, a complicated exact algorithm was proposed by Hartvigsen. Recently, a simple PTAS using local search was presented by Bosch-Calvo, Grandoni, and Ameli, but its validity proof is not easy. In this paper, we show a natural and simple decomposition theorem for triangle-free 2-matchings, which leads to a simpler validity proof of the PTAS for the problem.