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2013 IEEE 54th Annual Symposium on Foundations of Computer Science (2007)
Providence, Rhode Island
Oct. 21, 2007 to Oct. 23, 2007
ISSN: 0272-5428
ISBN: 0-7695-3010-9
pp: 702-712
<p>Linear and semidefinite programming are highly successful approaches for obtaining good approximations for NP-hard optimization problems. For example, breakthrough approximation algorithms for MAX CUT and SPARSEST CUT use semidefinite programming.</p><p> Perhaps the most prominent NP-hard problem whose exact approximation factor is still unresolved is VERTEX COVER. PCP-based techniques of Dinur and Safra [7] show that it is not possible to achieve a factor better than 1.36; on the other hand no known algorithm does better than the factor of 2 achieved by the simple greedy algorithm. Furthermore, there is a widespread belief that SDP techniques are the most promising methods available for improving upon this factor of 2.</p><p> Following a line of study initiated by Arora et al. [3], our aim is to show that a large family of LP and SDP based algorithms fail to produce an approximation for VERTEX COVER better than 2. Lovész and Schrijver [21] introduced the systems LS and LS_ + for systematically tightening LP and SDP relaxations, respectively, over many rounds. These systems naturally capture large classes of LP and SDP relaxations; indeed, LS_ + captures the celebrated SDP-based algorithms for MAX CUT and SPARSEST CUT mentioned above.</p><p> We rule out polynomial-time 2 - \Omega (1) approximations for VERTEX COVER using LS_ +. In particular, we prove an integrality gap of 2-o(1) for VERTEX COVER SDPs obtained by tightening the standard LP relaxation with \Omega (\sqrt {\log n/\log \log n} ) rounds of LS_ +. While tight integrality gaps were known for VERTEX COVER in the weaker LS system [23], previous results did not rule out a 2 - \Omega (1) approximation after even two rounds of LS_ +.</p>
Toniann Pitassi, Avner Magen, Iannis Tourlakis, Konstantinos Georgiou, "Integrality gaps of 2 - o(1) for Vertex Cover SDPs in the Lovész-Schrijver Hierarchy", 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, vol. 00, no. , pp. 702-712, 2007, doi:10.1109/FOCS.2007.35
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