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Approximating the independence number via the $$\vartheta$$-function. (English) Zbl 0895.90169
Summary: We describe an approximation algorithm for the independence number of a graph. If a graph on $$n$$ vertices has an independence number $$n/k+m$$ for some fixed integer $$k\geq 3$$ and some $$m>0$$, the algorithm finds, in random polynomial time, an independent set of size $$\widetilde{\Omega} (m^{3/(k+1)})$$, improving the best known previous algorithm of Boppana and Halldorsson that finds an independent set of size $$\Omega(m^{1/(k-1)})$$ in such a graph. The algorithm is based on semi-definite programming, some properties of the Lovász $$\vartheta$$-function of a graph and the recent algorithm of Karger, Motwani and Sudan for approximating the chromatic number of a graph. If the $$\vartheta$$-function of an $$n$$ vertex graph is at least $$Mn^{1-2/k}$$ for some absolute constant $$M$$, we describe another, related, efficient algorithm that finds an independent set of size $$k$$. Several examples show the limitations of the approach and the analysis together with some related arguments supply new results on the problem of estimating the largest possible ratio between the $$\vartheta$$-function and the independence number of a graph on $$n$$ vertices.

##### MSC:
 90C35 Programming involving graphs or networks
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