Abstract

Recently there has been extensive work on maintaining (approximate) maximum matchings in dynamic graphs. We consider a generalisation of this problem known as the maximum b-matching: Every node v has a positive integral capacity b_v, and the goal is to maintain an (approximate) maximum-cardinality subset of edges that contains at most b_v edges incident on every node v. The maximum matching problem is a special case of this problem where b_v = 1 for every node v. Bhattacharya, Henzinger and Italiano [ICALP 2015] showed how to maintain a O(1) approximate maximum b-matching in a graph in O(log^3 n) amortised update time. Their approximation ratio was a large (double digit) constant. We significantly improve their result both in terms of approximation ratio as well as update time. Specifically, we design a randomised dynamic algorithm that maintains a (2+epsilon)-approximate maximum $b$-matching in expected amortised O(1/epsilon^4) update time. Thus, for every constant epsilon in (0, 1), we get expected amortised O(1) update time. Our algorithm generalises the framework of Baswana, Gupta, Sen [FOCS 2011] and Solomon [FOCS 2016] for maintaining a maximal matching in a dynamic graph.

@InProceedings{bhattacharya_et_al:LIPIcs:2017:7844,
  author =	{Sayan Bhattacharya and Manoj Gupta and Divyarthi Mohan},
  title =	{{Improved Algorithm for Dynamic b-Matching}},
  booktitle =	{25th Annual European Symposium on Algorithms (ESA 2017)},
  pages =	{15:1--15:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-049-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{87},
  editor =	{Kirk Pruhs and Christian Sohler},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7844},
  URN =		{urn:nbn:de:0030-drops-78443},
  doi =		{10.4230/LIPIcs.ESA.2017.15},
  annote =	{Keywords: dynamic data structures, graph algorithms}
}