Finding a Maximum Independent Set in a Sparse Random Graph

Abstract

We consider the problem of finding a maximum independent set in a random graph. The random graph G is modelled as follows. Every edge is included independently with probability $\frac{d}{n}$, where d is some sufficiently large constant. Thereafter, for some constant α, a subset I of αn vertices is chosen at random, and all edges within this subset are removed. In this model, the planted independent set I is a good approximation for the maximum independent set I _max , but both I ∖ I _max and I _max ∖ I are likely to be nonempty. We present a polynomial time algorithms that with high probability (over the random choice of random graph G, and without being given the planted independent set I) finds a maximum independent set in G when $\alpha \geq \sqrt{c_0 \log d /d}$, where c ₀ is some sufficiently large constant independent of d.

This work was supported in part by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development. Part of this work was done while the authors were visiting Microsoft Research in Redmond, Washington.