A PTAS for Weighted Triangle-free 2-Matching

2026-03-09

AI Summary

This study addresses the triangle-free 2-matching problem in edge-weighted undirected graphs, which seeks a maximum-weight subgraph where each vertex has degree at most two and no three vertices form a triangle. This problem is known to be NP-hard. The paper presents the first polynomial-time approximation scheme (PTAS) for this problem, achieving a $(1-\varepsilon)$-approximation for any fixed constant $\varepsilon > 0$ in polynomial time. The approach leverages a local search strategy combined with a refined combinatorial structural analysis, thereby breaking through the long-standing $2/3$-approximation barrier. This advancement significantly enhances both the theoretical guarantees and practical applicability of algorithms for this fundamental graph optimization problem.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Bridging structured and unstructured data

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.

Problem

• Weighted Triangle-free 2-Matching
• 2-matching
• triangle-free
• approximation algorithm
• combinatorial optimization

Innovation

• PTAS
• Triangle-free 2-matching
• Local search
• Approximation algorithm
• Combinatorial optimization