On the Lovász Number of Certain Circulant Graphs
CIAC '00 Proceedings of the 4th Italian Conference on Algorithms and Complexity
Communication lower bounds via the chromatic number
FSTTCS'07 Proceedings of the 27th international conference on Foundations of software technology and theoretical computer science
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Approximating Semidefinite Packing Programs
SIAM Journal on Optimization
Product rules in semidefinite programming
FCT'07 Proceedings of the 16th international conference on Fundamentals of Computation Theory
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The /spl theta/ number of Lovasz and the /spl theta//sub 1/2/ of Schrijver, McEliece, Rodemich and Rumsey are convex (semidefinite) programming upper bounds on /spl alpha/(G), the size of a maximal independent set of G. It is known that /spl alpha/(G)/spl les//spl theta//sub 1/2/(G)/spl les//spl theta/(G)/spl les//spl chi/~(G), where /spl chi/~(G) is the clique cover number of G. The above inequalities suggest that perhaps /spl theta//sub 1/2/(G) approximates /spl alpha/(G) from above, and /spl theta/(G) approximates /spl chi/~(G) from below for every graph G. Can this approximation be to within a factor of at most n/sup 1-/spl epsiv// for some fixed /spl epsiv/0? We show, that the following three conjectures are equivalent: 1. /spl exist//spl epsiv/0 : /spl theta/(G) approximates /spl alpha/(G) for every G within a factor of n/sup n-/spl epsiv// 2. /spl exist//spl epsiv/0 : /spl theta/(G) approximates /spl chi/~(G) for every G within a factor of n/sup n-/spl epsiv// 3. /spl exist//spl epsiv/0 : /spl theta//sub 1/2/(G) approximates /spl alpha/(G) for every G within a factor of n/sup n-/spl epsiv// It is not impossible that /spl theta//sub 1/2/ approximates /spl chi/~(G), but the latter conjecture looks strictly stronger than 1-3. We give however a simple combinatorial reformulation of this one (we cannot find such for 1-3). We rule out some likely candidates for counterexamples to 1-3 by showing that /spl theta/(G) approximates /spl alpha/(G) and /spl chi/~(G) for those graphs G that come from the Hamming scheme.