The Matching Augmentation Problem: A $\frac{7}{4}$-Approximation Algorithm

Abstract

We present a $\frac{7}{4}$ 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 $\frac{7}{4}$ 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).

[v2]: based on a thorough journal review, the submission has been revised to improve the exposition, though the results and proofs are the same (modulo expository improvements); there are several changes, mostly in sections 4, 5, 6; more informal discussion (of the credit scheme) has been added in sections 5.2 and 5.3; nevertheless, sections 4.4 and 5.3 need patience and effort