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DOI: 10.4230/LIPIcs.STACS.2011.519
URN: urn:nbn:de:0030-drops-30402
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2011/3040/
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Babenko, Maxim ; Gusakov, Alexey

New Exact and Approximation Algorithms for the Star Packing Problem in Undirected Graphs

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Abstract

By a T-star we mean a complete bipartite graph K_{1,t} for some t <= T. For an undirected graph G, a T-star packing is a collection of node-disjoint T-stars in G.
For example, we get ordinary matchings for $T = 1$ and packings of paths of length 1 and 2 for $T = 2$. Hereinafter we assume that T >= 2.

Hell and Kirkpatrick devised an ad-hoc augmenting algorithm that finds a T-star packing covering the maximum number of nodes. The latter algorithm also yields a min-max formula.

We show that T-star packings are reducible to network flows, hence the above problem is solvable in $O(m sqrt(n))$ time (hereinafter n denotes the number of nodes in G, and m --- the number of edges).

For the edge-weighted case (in which weights may be assumed positive) finding a maximum $T$-packing is NP-hard. A novel 9/4 T/(T + 1)-factor approximation algorithm is presented.

For non-negative node weights the problem reduces to a special case of a max-cost flow. We develop a divide-and-conquer approach that solves it in O(m sqrt(n) log(n)) time. The node-weighted problem with arbitrary weights is more difficult. We prove that it is NP-hard for T >= 3 and is solvable in strongly-polynomial time for T = 2.

BibTeX - Entry

@InProceedings{babenko_et_al:LIPIcs:2011:3040,
  author =	{Maxim Babenko and Alexey Gusakov},
  title =	{{New Exact and Approximation Algorithms for the Star Packing Problem in Undirected Graphs}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) },
  pages =	{519--530},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Thomas Schwentick and Christoph D{\"u}rr},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2011/3040},
  URN =		{urn:nbn:de:0030-drops-30402},
  doi =		{10.4230/LIPIcs.STACS.2011.519},
  annote =	{Keywords: graph algorithms, approximation algorithms, generalized matchings, flows, weighted packings}
}

Keywords: graph algorithms, approximation algorithms, generalized matchings, flows, weighted packings
Collection: 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)
Issue Date: 2011
Date of publication: 11.03.2011

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