eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-08-27
74:1
74:16
10.4230/LIPIcs.MFCS.2018.74
article
A Simple Augmentation Method for Matchings with Applications to Streaming Algorithms
Konrad, Christian
1
Department of Computer Science, University of Bristol, Merchant Venturers Building, Woodland Road, BS8 1UB, United Kingdom
Given a graph G, it is well known that any maximal matching M in G is at least half the size of a maximum matching M^*. In this paper, we show that if G is bipartite, then running the Greedy matching algorithm on a sampled subgraph of G produces enough additional edges that can be used to augment M such that the resulting matching is of size at least (2 - sqrt{2})|M^*| ~~ 0.5857 |M^*| (ignoring lower order terms) with high probability.
The main applications of our method lie in the area of data streaming algorithms, where an algorithm performs few passes over the edges of an n-vertex graph while maintaining a memory of size O(n polylog n). Our method immediately yields a very simple two-pass algorithm for Maximum Bipartite Matching (MBM) with approximation factor 0.5857, which only runs the Greedy matching algorithm in each pass. This slightly improves on the much more involved 0.583-approximation algorithm of Esfandiari et al. [ICDMW 2016]. To obtain our main result, we combine our method with a residual sparsity property of the random order Greedy algorithm and give a one-pass random order streaming algorithm for MBM with approximation factor 0.5395. This substantially improves upon the one-pass random order 0.505-approximation algorithm of Konrad et al. [APPROX 2012].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol117-mfcs2018/LIPIcs.MFCS.2018.74/LIPIcs.MFCS.2018.74.pdf
Matchings
augmenting paths
streaming algorithms
random order