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DOI: 10.4230/LIPIcs.STACS.2011.129
URN: urn:nbn:de:0030-drops-30051
URL: http://drops.dagstuhl.de/opus/volltexte/2011/3005/
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Kolman, Petr ; Scheideler, Christian

Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing

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Abstract

An elementary h-route flow, for an integer h >= 1, is a set of h edge-disjoint paths between a source and a sink, each path carrying a unit of flow, and an h-route flow is a non-negative linear combination of elementary h-route flows. An h-route cut is a set of edges whose removal decreases the maximum h-route flow between a given source-sink pair (or between every source-sink pair in the multicommodity setting) to zero. The main result of this paper is an approximate duality theorem for multicommodity $h$-route cuts and flows, for h <= 3: The size of a minimum h-route cut is at least f/h and at most O(log^3(k)f) where f is the size of the maximum h-route flow and k is the number of commodities. The main step towards the proof of this duality is the design and analysis of a polynomial-time approximation algorithm for the minimum h-route cut problem for h=3 that has an approximation ratio of O(log^3 k). Previously, polylogarithmic approximation was known only for $h$-route cuts for h <= 2. A key ingredient of our algorithm is a novel rounding technique that we call multilevel ball-growing. Though the proof of the duality relies on this algorithm, it is not a straightforward corollary of it as in the case of classical multicommodity flows and cuts. Similar results are shown also for the sparsest multiroute cut problem.

BibTeX - Entry

@InProceedings{kolman_et_al:LIPIcs:2011:3005,
  author =	{Petr Kolman and Christian Scheideler},
  title =	{{Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) },
  pages =	{129--140},
  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/2011/3005},
  URN =		{urn:nbn:de:0030-drops-30051},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2011.129},
  annote =	{Keywords: Multicommodity flow, Multiroute flow, Cuts, Duality}
}

Keywords: Multicommodity flow, Multiroute flow, Cuts, Duality
Seminar: 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)
Issue Date: 2011
Date of publication: 11.03.2011


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