The Probable Value of the Lovász--Schrijver Relaxations for Maximum Independent Set

Lovász and Schrijver [LS91] 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 Lovász theta function [Lov79]. We show that on a random graph Gn,1/2, the value of the rth relaxation in the sequence is roughly √ n/2r, 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 = Θ(logn) relaxation in the sequence.