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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.21
URN: urn:nbn:de:0030-drops-75705
URL: http://drops.dagstuhl.de/opus/volltexte/2017/7570/
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Rabani, Yuval ; Venkat, Rakesh

Approximating Sparsest Cut in Low Rank Graphs via Embeddings from Approximately Low Dimensional Spaces

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Abstract

We consider the problem of embedding a finite set of points x_1, ... , x_n in R^d that satisfy l_2^2 triangle inequalities into l_1, when the points are approximately low-dimensional. Goemans (unpublished, appears in a work of Magen and Moharammi (2008) ) showed that such points residing in exactly d dimensions can be embedded into l_1 with distortion at most sqrt{d}. We prove the following robust analogue of this statement: if there exists a r-dimensional subspace Pi such that the projections onto this subspace satisfy sum_{i,j in [n]} norm{Pi x_i - Pi x_j}_2^2 >= Omega(1) * sum_{i,j \in [n]} norm{x_i - x_j}_2^2, then there is an embedding of the points into l_1 with O(sqrt{r}) average distortion. A consequence of this result is that the integrality gap of the well-known Goemans-Linial SDP relaxation for the Uniform Sparsest Cut problem is O(sqrt{r}) on graphs G whose r-th smallest normalized eigenvalue of the Laplacian satisfies lambda_r(G)/n >= Omega(1)*Phi_{SDP}(G). Our result improves upon the previously known bound of O(r) on the average distortion, and the integrality gap of the Goemans-Linial SDP under the same preconditions, proven in [Deshpande and Venkat, 2014], and [Deshpande, Harsha and Venkat 2016].

BibTeX - Entry

@InProceedings{rabani_et_al:LIPIcs:2017:7570,
  author =	{Yuval Rabani and Rakesh Venkat},
  title =	{{Approximating Sparsest Cut in Low Rank Graphs via Embeddings from Approximately Low Dimensional Spaces}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{21:1--21:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Klaus Jansen and Jos{\'e} D. P. Rolim and David Williamson and Santosh S. Vempala},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7570},
  URN =		{urn:nbn:de:0030-drops-75705},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.21},
  annote =	{Keywords: Metric Embeddings, Sparsest Cut, Negative type metrics, Approximation Algorithms}
}

Keywords: Metric Embeddings, Sparsest Cut, Negative type metrics, Approximation Algorithms
Seminar: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)
Issue Date: 2017
Date of publication: 31.07.2017


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