The Matching Augmentation Problem: A $\frac74$-Approximation Algorithm

TL;DR

This paper introduces a 7/4-approximation algorithm for the matching augmentation problem, improving the efficiency of finding minimum-cost 2-edge connected spanning subgraphs in weighted multi-graphs.

Contribution

It provides a novel reduction technique for MAP instances and a new approximation algorithm with a proven 7/4 guarantee for well-structured cases.

Findings

Achieves a 7/4 approximation ratio for MAP

Reduces general MAP instances to well-structured cases

Uses ear-augmentation and min-cost 2-edge cover techniques

Abstract

We present a $\frac74$ approximation algorithm for the matching augmentation problem (MAP): given a multi-graph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost. We first present a reduction of any given MAP instance to a collection of well-structured MAP instances such that the approximation guarantee is preserved. Then we present a $\frac74$ approximation algorithm for a well-structured MAP instance. The algorithm starts with a min-cost 2-edge cover and then applies ear-augmentation steps. We analyze the cost of the ear-augmentations using an approach similar to the one proposed by Vempala and Vetta for the (unweighted) min-size 2-ECSS problem (`Factor 4/3 approximations for minimum 2-connected subgraphs,' APPROX 2000, LNCS 1913, pp.262-273).