Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yusuke Kobayashi, Takashi Noguchi

TL;DR

A natural and simple decomposition theorem for triangle-free 2-matchings is shown, which leads to a simpler validity proof of the PTAS for the problem of maximum cardinality triangle-free 2-matching in a given graph.

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.

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

TL;DR

A natural and simple decomposition theorem for triangle-free 2-matchings is shown, which leads to a simpler validity proof of the PTAS for the problem of maximum cardinality triangle-free 2-matching in a given graph.

Abstract

A triangle-free (simple) 2-matching is an edge set that has at most edges incident to each vertex and contains no cycle of length . 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.

Paper Structure

This paper contains 10 sections, 5 theorems, 4 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Study
  3. Preliminary
  4. Proof of Theorem \ref{['thm:tri-part']}
  5. Case 1.
  6. Case 2.
  7. Case 3.
  8. Case 3'.
  9. Case 4.
  10. Concluding Remarks

Key Result

Theorem 1

Let $G$ be a simple graph, and $A_1$ and $A_2$ be triangle-free 2-matchings in $G$. Then there is a partition $\mathcal{P}$ of $A_1\bigtriangleup A_2$ into alternating trails w.r.t. $(A_1, A_2)$ such that $A_i\bigtriangleup P$ is a triangle-free 2-matching for $i=1, 2$ and for any $P\in \mathcal{P}$

Theorems & Definitions (10)

  • Theorem 1: Decomposition for triangle-free 2-matchings
  • Corollary 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4: Decomposition for $\mathcal{T}$-free 2-matchings
  • proof
  • Claim 1
  • proof : Proof of the Claim
  • Corollary 5