A PTAS for Weighted Triangle-free 2-Matching

TL;DR

This paper presents a PTAS for WTF2M, i.e., a polynomial-time $(1-\varepsilon)$-approximation algorithm for any given constant $\varepsilon>0$.

Abstract

In the Weighted Triangle-Free 2-Matching problem (WTF2M), we are given an undirected edge-weighted graph. Our goal is to compute a maximum-weight subgraph that is a 2-matching (i.e., no node has degree more than $2$) and triangle-free (i.e., it does not contain any cycle with $3$ edges). One of the main motivations for this and related problems is their practical and theoretical connection with the Traveling Salesperson Problem and with some $2$-connectivity network design problems. WTF2M is not known to be NP-hard and at the same time no polynomial-time algorithm to solve it is known in the general case (polynomial-time algorithms are known only for some special cases). The best-known (folklore) approximation algorithm for this problem simply computes a maximum-weight 2-matching, and then drops the cheapest edge of each triangle: this gives a $2/3$ approximation. In this paper we present a PTAS for WTF2M, i.e., a polynomial-time $(1-\varepsilon)$-approximation algorithm for any given constant $\varepsilon>0$. Our result is based on a simple local-search algorithm and a non-trivial analysis.

Figures (9)

• Figure 1: Reduction for Case \ref{['case:A1A2']}.
• Figure 2: Reduction for Case \ref{['case:2A1A2loop']}
• Figure 3: Reduction for Case \ref{['case:2A1A2:red']}.
• Figure 4: Reduction for Case \ref{['case:A12A2loop']}
• Figure 5: Reduction for Case \ref{['case:A12A2:red']}.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• proof
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['thr:ptas']}
• Theorem 3
• Lemma 2
• proof
• proof