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.CCC.2019.8
URN: urn:nbn:de:0030-drops-108309
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10830/
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O'Donnell, Ryan ; Schramm, Tselil

Sherali - Adams Strikes Back

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LIPIcs-CCC-2019-8.pdf (0.7 MB)

Abstract

Let G be any n-vertex graph whose random walk matrix has its nontrivial eigenvalues bounded in magnitude by 1/sqrt{Delta} (for example, a random graph G of average degree Theta(Delta) typically has this property). We show that the exp(c (log n)/(log Delta))-round Sherali - Adams linear programming hierarchy certifies that the maximum cut in such a G is at most 50.1 % (in fact, at most 1/2 + 2^{-Omega(c)}). For example, in random graphs with n^{1.01} edges, O(1) rounds suffice; in random graphs with n * polylog(n) edges, n^{O(1/log log n)} = n^{o(1)} rounds suffice.
Our results stand in contrast to the conventional beliefs that linear programming hierarchies perform poorly for max-cut and other CSPs, and that eigenvalue/SDP methods are needed for effective refutation. Indeed, our results imply that constant-round Sherali - Adams can strongly refute random Boolean k-CSP instances with n^{ceil[k/2] + delta} constraints; previously this had only been done with spectral algorithms or the SOS SDP hierarchy.

BibTeX - Entry

@InProceedings{odonnell_et_al:LIPIcs:2019:10830,
  author =	{Ryan O'Donnell and Tselil Schramm},
  title =	{{Sherali - Adams Strikes Back}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{8:1--8:30},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Amir Shpilka},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10830},
  URN =		{urn:nbn:de:0030-drops-108309},
  doi =		{10.4230/LIPIcs.CCC.2019.8},
  annote =	{Keywords: Linear programming, Sherali, Adams, max-cut, graph eigenvalues, Sum-of-Squares}
}

Keywords: Linear programming, Sherali, Adams, max-cut, graph eigenvalues, Sum-of-Squares
Collection: 34th Computational Complexity Conference (CCC 2019)
Issue Date: 2019
Date of publication: 16.07.2019

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