Tighter Linear and Semidefinite Relaxations for Max-cut Based on the Lovász–schrijver Lift-and-project Procedure∗

We study how the lift-and-project method introduced by Lovász and Schrijver [SIAM J. Optim., 1 (1991), pp. 166–190] applies to the cut polytope. We show that the cut polytope of a graph can be found in k iterations if there exist k edges whose contraction produces a graph with no K5-minor. Therefore, for a graph G with n ≥ 4 nodes with stability number α(G), n − 4 iterations suffice instead of the m (number of edges) iterations required in general and, under some assumption, n−α(G)− 3 iterations suffice. The exact number of needed iterations is determined for small n ≤ 7 by a detailed analysis of the new relaxations. If positive semidefiniteness is added to the construction, then one finds in one iteration a relaxation of the cut polytope which is tighter than its basic semidefinite relaxation and than another one introduced recently by Anjos and Wolkowicz [Discrete Appl. Math., to appear]. We also show how the Lovász–Schrijver relaxations for the stable set polytope of G can be strengthened using the corresponding relaxations for the cut polytope of the graph G∇ obtained from G by adding a node adjacent to all nodes of G.