A Simple LP-Based Approximation Algorithm for the Matching Augmentation Problem

TL;DR

This paper introduces a straightforward LP-guided algorithm for the Matching Augmentation Problem that improves approximation ratios and provides new bounds on the LP relaxation's integrality gap.

Contribution

The paper presents a simple LP-based approximation algorithm for MAP that outperforms previous methods and offers tighter bounds on the LP relaxation.

Findings

Algorithm achieves better than 2-approximation ratio.

Provides an improved upper bound on the LP relaxation's integrality gap.

Uses properties of extreme point solutions for analysis.

Abstract

The Matching Augmentation Problem (MAP) has recently received significant attention as an important step towards better approximation algorithms for finding cheap $2$-edge connected subgraphs. This has culminated in a $\frac{5}{3}$-approximation algorithm. However, the algorithm and its analysis are fairly involved and do not compare against the problem's well-known LP relaxation called the cut LP. In this paper, we propose a simple algorithm that, guided by an optimal solution to the cut LP, first selects a DFS tree and then finds a solution to MAP by computing an optimum augmentation of this tree. Using properties of extreme point solutions, we show that our algorithm always returns (in polynomial time) a better than $2$-approximation when compared to the cut LP. We thereby also obtain an improved upper bound on the integrality gap of this natural relaxation.