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Documents authored by Magen, Avner

Document
DOI: 10.4230/LIPIcs.FSTTCS.2011.41
Tight Gaps for Vertex Cover in the Sherali-Adams SDP Hierarchy

Authors: Siavosh Benabbas, Siu On Chan, Konstantinos Georgiou, and Avner Magen

Published in: LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

Abstract
We give the first tight integrality gap for Vertex Cover in the Sherali-Adams SDP system. More precisely, we show that for every \epsilon >0, the standard SDP for Vertex Cover that is strengthened with the level-6 Sherali-Adams system has integrality gap 2-\epsilon. To the best of our knowledge this is the first nontrivial tight integrality gap for the Sherali-Adams SDP hierarchy for a combinatorial problem with hard constraints. For our proof we introduce a new tool to establish Local-Global Discrepancy which uses simple facts from high-dimensional geometry. This allows us to give Sherali-Adams solutions with objective value n(1/2+o(1)) for graphs with small (2+o(1)) vector chromatic number. Since such graphs with no linear size independent sets exist, this immediately gives a tight integrality gap for the Sherali-Adams system for superconstant number of tightenings. In order to obtain a Sherali-Adams solution that also satisfies semidefinite conditions, we reduce semidefiniteness to a condition on the Taylor expansion of a reasonably simple function that we are able to establish up to constant-level SDP tightenings. We conjecture that this condition holds even for superconstant levels which would imply that in fact our solution is valid for superconstant level Sherali-Adams SDPs.
Cite as

Siavosh Benabbas, Siu On Chan, Konstantinos Georgiou, and Avner Magen. Tight Gaps for Vertex Cover in the Sherali-Adams SDP Hierarchy. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 41-54, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{benabbas_et_al:LIPIcs.FSTTCS.2011.41,
  author =	{Benabbas, Siavosh and Chan, Siu On and Georgiou, Konstantinos and Magen, Avner},
  title =	{{Tight Gaps for Vertex Cover in the Sherali-Adams SDP Hierarchy}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
  pages =	{41--54},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-34-7},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{13},
  editor =	{Chakraborty, Supratik and Kumar, Amit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.41},
  URN =		{urn:nbn:de:0030-drops-33299},
  doi =		{10.4230/LIPIcs.FSTTCS.2011.41},
  annote =	{Keywords: Vertex Cover, Integrality Gap, Lift-and-Project systems, Linear Programming, Semidefinite Programming}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2009.2319
On the Tightening of the Standard SDP for Vertex Cover with $ell_1$ Inequalities

Authors: Konstantinos Georgiou, Avner Magen, and Iannis Tourlakis

Published in: LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

Abstract
We show that the integrality gap of the standard SDP for \vc~on instances of $n$ vertices remains $2-o(1)$ even after the addition of \emph{all} hypermetric inequalities. Our lower bound requires new insights into the structure of SDP solutions behaving like $\ell_1$ metric spaces when one point is removed. We also show that the addition of all $\ell_1$ inequalities eliminates any solutions that are not convex combination of integral solutions. Consequently, we provide the strongest possible separation between hypermetrics and $\ell_1$ inequalities with respect to the tightening of the standard SDP for \vc.
Cite as

Konstantinos Georgiou, Avner Magen, and Iannis Tourlakis. On the Tightening of the Standard SDP for Vertex Cover with $ell_1$ Inequalities. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 203-214, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{georgiou_et_al:LIPIcs.FSTTCS.2009.2319,
  author =	{Georgiou, Konstantinos and Magen, Avner and Tourlakis, Iannis},
  title =	{{On the Tightening of the Standard SDP for Vertex Cover with \$ell\underline1\$ Inequalities}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{203--214},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-13-2},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{4},
  editor =	{Kannan, Ravi and Narayan Kumar, K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2319},
  URN =		{urn:nbn:de:0030-drops-23195},
  doi =		{10.4230/LIPIcs.FSTTCS.2009.2319},
  annote =	{Keywords: Semidefinite Programming, Vertex Cover, Integrality Gap, Hypermetric Inequalities}
}
Document
DOI: 10.4230/DagSemProc.05291.4
Sublinear Geometric Algorithms

Authors: Bernard Chazelle, Ding Liu, and Avner Magen

Published in: Dagstuhl Seminar Proceedings, Volume 5291, Sublinear Algorithms (2006)

Abstract
We present sublinear algorithms to such problems as Detecting of Polytope intersection, Shortest Path on 3D convex Polytopes and volume approximation.
Cite as

Bernard Chazelle, Ding Liu, and Avner Magen. Sublinear Geometric Algorithms. In Sublinear Algorithms. Dagstuhl Seminar Proceedings, Volume 5291, pp. 1-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{chazelle_et_al:DagSemProc.05291.4,
  author =	{Chazelle, Bernard and Liu, Ding and Magen, Avner},
  title =	{{Sublinear Geometric Algorithms}},
  booktitle =	{Sublinear Algorithms},
  pages =	{1--18},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{5291},
  editor =	{Artur Czumaj and S. Muthu Muthukrishnan and Ronitt Rubinfeld and Christian Sohler},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05291.4},
  URN =		{urn:nbn:de:0030-drops-5548},
  doi =		{10.4230/DagSemProc.05291.4},
  annote =	{Keywords: Sublinear algorithms, computational geometry}
}
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