Matching Augmentation via Simultaneous Contractions

TL;DR

This paper introduces a polynomial-time approximation algorithm for the matching augmentation problem that improves the approximation ratio from 1.666 to 1.625 by using novel contraction techniques and structural reductions.

Contribution

It presents a new approximation algorithm with a ratio of 1.625 for MAP, utilizing simultaneous contractions and structural reductions to improve previous bounds.

Findings

Achieved a 1.625-approximation ratio for MAP

Developed a reduction preserving approximation ratio for structured instances

Provided improved lower bounds for non-contractible instances

Abstract

We consider the matching augmentation problem (MAP), where a matching of a graph needs to be extended into a $2$-edge-connected spanning subgraph by adding the minimum number of edges to it. We present a polynomial-time algorithm with an approximation ratio of $13/8 = 1.625$ improving upon an earlier $5/3$-approximation. The improvement builds on a new $\alpha$-approximation preserving reduction for any $\alpha\geq 3/2$ from arbitrary MAP instances to well-structured instances that do not contain certain forbidden structures like parallel edges, small separators, and contractible subgraphs. We further introduce, as key ingredients, the technique of repeated simultaneous contractions and provide improved lower bounds for instances that cannot be contracted.