2012 IEEE 27th Conference on Computational Complexity (2007)

San Diego, California

June 13, 2007 to Mar. 16, 2007

ISSN: 1093-0159

ISBN: 0-7695-2780-9

pp: 205-216

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/CCC.2007.2

Luca Trevisan , UC Berkeley, USA

Madhur Tulsiani , UC Berkeley, USA

Grant Schoenebeck , UC Berkeley, USA

ABSTRACT

We study semidefinite programming relaxations of Vertex Cover arising from repeated applications of the LS+ "lift-and-project" method of Lovasz and Schrijver starting from the standard linear programming relaxation. <p>Goemans and Kleinberg prove that after one round of LS+ the integrality gap remains arbitrarily close to 2. Charikar proves an integrality gap of 2, later strengthened by Hatami, Magen, and Markakis, for stronger relaxations that are, however, incomparable with two rounds of LS+. Subsequent work by Georgiou, Magen, Pitassi, and Tourlakis shows that the integrality gap remains 2 - \varepsilon after \Omega(\sqrt {\frac{{\log n}} {{\log \log n}}} ) rounds [?].</p> <p>We prove that the integrality gap remains at least 7/6 - \varepsilon after c_{\varepsilon}n rounds, where n is the number of vertices and c_\varepsilon \ge 0 is a constant that depends only on \varepsilon.</p>

INDEX TERMS

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CITATION

Luca Trevisan,
Madhur Tulsiani,
Grant Schoenebeck,
"A Linear Round Lower Bound for Lovasz-Schrijver SDP Relaxations of Vertex Cover",

*2012 IEEE 27th Conference on Computational Complexity*, vol. 00, no. , pp. 205-216, 2007, doi:10.1109/CCC.2007.2