A comparison of the Sherali-Adams, Lovász-Schrijver and Lasserre relaxations for 0-1 programming

Sherali and Adams [SA90], Lov'asz and Schrijver [LS91] and, recently, Lasserre [Las01b] have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a $0-1$ polytope $Psubseteq oR^n$ converging to $P$ in $n$ steps. Lasserre's approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments. We present the three methods in a common elementary framework and show that the Lasserre construction provides the tightest relaxations of $P$. As an application this gives a direct simple proof for the convergence of the Lasserre's hierarchy. We describe applications to the stable set polytope and to the cut polytope.