The probable value of the Lovasz-Schrijver relaxations for maximum independent set

Lovasz and Schrijver [SIAM J. Optim., 1 (1991), pp. 166-190] devised a lift-and-project method that produces a sequence of convex relaxations for the problem of finding in a graph an independent set ( or a clique) of maximum size. Each relaxation in the sequence is tighter than the one before it, while the first relaxation is already at least as strong as the Lovasz theta function [IEEE Trans. Inform. Theory, 25 (1979), pp. 1-7]. We show that on a random graph G(n,1/2), the value of the rth relaxation in the sequence is roughly rootn/2(r), almost surely. It follows that for those relaxations known to be efficiently computable, namely, for r=O(1), the value of the relaxation is comparable to the theta function. Furthermore, a perfectly tight relaxation is almost surely obtained only at the r=Theta(log n) relaxation in the sequence.