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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.738
URN: urn:nbn:de:0030-drops-53334
URL: http://drops.dagstuhl.de/opus/volltexte/2015/5333/
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Dinitz, Michael ; Krauthgamer, Robert ; Wagner, Tal

Towards Resistance Sparsifiers

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Abstract

We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a (1+epsilon)-resistance sparsifier of size ~O(n/epsilon), and conjecture this bound holds for all graphs on n nodes. In comparison, spectral sparsification is a strictly stronger notion and requires Omega(n/epsilon^2) edges even on the complete graph. Our approach leads to the following structural question on graphs: Does every dense regular expander contain a sparse regular expander as a subgraph? Our main technical contribution, which may of independent interest, is a positive answer to this question in a certain setting of parameters. Combining this with a recent result of von Luxburg, Radl, and Hein (JMLR, 2014) leads to the aforementioned resistance sparsifiers.

BibTeX - Entry

@InProceedings{dinitz_et_al:LIPIcs:2015:5333,
  author =	{Michael Dinitz and Robert Krauthgamer and Tal Wagner},
  title =	{{Towards Resistance Sparsifiers}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{738--755},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Naveen Garg and Klaus Jansen and Anup Rao and Jos{\'e} D. P. Rolim},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5333},
  URN =		{urn:nbn:de:0030-drops-53334},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.738},
  annote =	{Keywords: edge sparsification, spectral sparsifier, graph expansion, effective resistance, commute time}
}

Keywords: edge sparsification, spectral sparsifier, graph expansion, effective resistance, commute time
Seminar: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)
Issue Date: 2015
Date of publication: 28.07.2015


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