License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.34
URN: urn:nbn:de:0030-drops-46865
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2014/4686/
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Alon, Noga ; Lee, Troy ; Shraibman, Adi

The Cover Number of a Matrix and its Algorithmic Applications

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Abstract

Given a matrix A, we study how many epsilon-cubes are required to cover the convex hull of the columns of A. We show bounds on this cover number in terms of VC dimension and the gamma_2 norm and give algorithms for enumerating elements of a cover. This leads to algorithms for computing approximate Nash equilibria that unify and extend several previous results in the literature. Moreover, our approximation algorithms can be applied quite generally to a family of quadratic optimization problems that also includes finding the k-by-k combinatorial rectangle of a matrix. In particular, for this problem we give the first quasi-polynomial time additive approximation algorithm that works for any matrix A in [0,1]^{m x n}.

BibTeX - Entry

@InProceedings{alon_et_al:LIPIcs:2014:4686,
  author =	{Noga Alon and Troy Lee and Adi Shraibman},
  title =	{{The Cover Number of a Matrix and its Algorithmic Applications}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{34--47},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Klaus Jansen and Jos{\'e} D. P. Rolim and Nikhil R. Devanur and Cristopher Moore},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2014/4686},
  URN =		{urn:nbn:de:0030-drops-46865},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.34},
  annote =	{Keywords: Approximation algorithms, Approximate Nash equilibria, Cover number, VC dimension}
}

Keywords: Approximation algorithms, Approximate Nash equilibria, Cover number, VC dimension
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)
Issue Date: 2014
Date of publication: 04.09.2014

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