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Gap, cosum, and product properties of the $\theta'$ bound on the clique number

Immanuel Bomze (immanuel.bomze***at***univie.ac.at)
Florian Frommlet (florian.frommlet***at***univie.ac.at)
Marco Locatelli (locatelli***at***di.unito.it])

Abstract: In a paper published 1978, McEliece, Rodemich and Rumsey improved Lov\'asz' bound for the Maximum Clique Problem. This strengthening has become well-known under the name Lov\'asz-Schrijver bound and is usually denoted by $\theta'$. This article now deals with situations where this bound is not exact. To provide instances for which the gap between this bound and the actual clique number can be arbitrarily large, we establish homomorphy results for this bound under cosums and products of graphs, and study the special class of circulant graphs: here we show that for circulant graphs of prime order there must be a positive gap between the clique number and the bound.

Keywords: circulant graph, direct graph product, lexicographic graph product, maximum clique problem, semidefinite programming

Category 1: Combinatorial Optimization (Graphs and Matroids )

Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Citation: Technical Report TR-ISDS {\bf 2007-06}, University of Vienna, to appear in: Optimization (2010).

Download: [PDF]

Entry Submitted: 04/24/2007
Entry Accepted: 04/25/2007
Entry Last Modified: 03/08/2010

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