When quoting this document, please refer to the following
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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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 Goemans (unpublished, appears in a work of Magen and Moharammi (2008) ) showed that such points residing in exactly d dimensions can be embedded into l_1 with distortion at most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},