Rabani, Yuval ;
Venkat, Rakesh
Approximating Sparsest Cut in Low Rank Graphs via Embeddings from Approximately Low Dimensional Spaces
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 lowdimensional. Goemans (unpublished, appears in a work of Magen and Moharammi (2008) ) showed that such points residing in exactly d dimensions can be embedded into l_1 with distortion at most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 rdimensional 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 wellknown GoemansLinial SDP relaxation for the Uniform Sparsest Cut problem is O(sqrt{r}) on graphs G whose rth 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 GoemansLinial 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 = {21:121:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770446},
ISSN = {18688969},
year = {2017},
volume = {81},
editor = {Klaus Jansen and Jos{\'e} D. P. Rolim and David Williamson and Santosh S. Vempala},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7570},
URN = {urn:nbn:de:0030drops75705},
doi = {10.4230/LIPIcs.APPROXRANDOM.2017.21},
annote = {Keywords: Metric Embeddings, Sparsest Cut, Negative type metrics, Approximation Algorithms}
}
11.08.2017
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: 

11.08.2017 