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DOI: 10.4230/LIPIcs.ESA.2019.71
URN: urn:nbn:de:0030-drops-111926
URL: https://drops.dagstuhl.de/opus/volltexte/2019/11192/
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Li, Huan ; Sun, He ; Zanetti, Luca

Hermitian Laplacians and a Cheeger Inequality for the Max-2-Lin Problem

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LIPIcs-ESA-2019-71.pdf (0.6 MB)


Abstract

We study spectral approaches for the MAX-2-LIN(k) problem, in which we are given a system of m linear equations of the form x_i - x_j is equivalent to c_{ij} mod k, and required to find an assignment to the n variables {x_i} that maximises the total number of satisfied equations. We consider Hermitian Laplacians related to this problem, and prove a Cheeger inequality that relates the smallest eigenvalue of a Hermitian Laplacian to the maximum number of satisfied equations of a MAX-2-LIN(k) instance I. We develop an O~(kn^2) time algorithm that, for any (1-epsilon)-satisfiable instance, produces an assignment satisfying a (1 - O(k)sqrt{epsilon})-fraction of equations. We also present a subquadratic-time algorithm that, when the graph associated with I is an expander, produces an assignment satisfying a (1- O(k^2)epsilon)-fraction of the equations. Our Cheeger inequality and first algorithm can be seen as generalisations of the Cheeger inequality and algorithm for MAX-CUT developed by Trevisan.

BibTeX - Entry

@InProceedings{li_et_al:LIPIcs:2019:11192,
  author =	{Huan Li and He Sun and Luca Zanetti},
  title =	{{Hermitian Laplacians and a Cheeger Inequality for the Max-2-Lin Problem}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{71:1--71:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Michael A. Bender and Ola Svensson and Grzegorz Herman},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/11192},
  URN =		{urn:nbn:de:0030-drops-111926},
  doi =		{10.4230/LIPIcs.ESA.2019.71},
  annote =	{Keywords: Spectral methods, Hermitian Laplacians, the Max-2-Lin problem, Unique Games}
}

Keywords: Spectral methods, Hermitian Laplacians, the Max-2-Lin problem, Unique Games
Seminar: 27th Annual European Symposium on Algorithms (ESA 2019)
Issue Date: 2019
Date of publication: 06.09.2019


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