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 (in: Jansen and Khuller (eds.) Approximation Algorithms for Combinatorial Optimization, Third International Workshop, APPROX 2000, Proceedings, LNCS 1913, pp.262–273, Springer, Berlin, 2000).
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Notes
Throughout, we abuse the term ear; although Y is not an ear, one may view the minimal subpath of Y from U to R, call it \(Y'\), as an ear of G w.r.t. \(C_0\), i.e. \(Y'\) is a path of G that has both end nodes in \(C_0\) and has no internal node in \(C_0\).
If \(s_0=t_0\), then note that \(s_0\) is incident to \(\ge 2\) bridges of \(C = C - E(P(s_0,t_0))\), hence, we can ensure that \(t_1\not =s_1\).
From this sentence till the end of Sect. 6, “node” means a node of G (and not a node of \({\hat{G}}\)).
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Acknowledgements
We are grateful to several colleagues for their careful reading of preliminary drafts and for their comments. We thank an anonymous reviewer for a thorough review. J.Cheriyan acknowledges support from the Natural Sciences & Engineering Research Council of Canada (NSERC), No. RGPIN–2014–04351. F.Grandoni is partially supported by the SNSF Grants 200021_159697/1 and 200020B_182865/1.
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Cheriyan, J., Dippel, J., Grandoni, F. et al. The matching augmentation problem: a \(\frac{7}{4}\)-approximation algorithm. Math. Program. 182, 315–354 (2020). https://doi.org/10.1007/s10107-019-01394-z
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DOI: https://doi.org/10.1007/s10107-019-01394-z
Keywords
- 2-Edge connected graph
- 2-Edge covers
- Approximation algorithms
- Bridges
- Connectivity augmentation
- Forest augmentation problem
- Matching augmentation problem
- Network design