Summary: Approximating the independence number via the -function
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 3 and
some m > 0, the algorithm finds, in random polynomial time, an independent set of size
), improving the best known previous algorithm of Boppana and Halldorsson that
finds an independent set of size (m1/(k-1)
) in such a graph. The algorithm is based on semi-
definite programming, some properties of the Lov´asz -function of a graph and the recent
algorithm of Karger, Motwani and Sudan for approximating the chromatic number of a graph.
If the -function of an n vertex graph is at least Mn1-2/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 -function and the independence number of a graph on n vertices.