Exponential Lower Bounds on the Lengths of Some Classes of Branch-and-Cut Proofs
Mathematics of Operations Research
Discrete Applied Mathematics
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Modeling, optimization and computation for software verification
HSCC'05 Proceedings of the 8th international conference on Hybrid Systems: computation and control
The equivalence of semidefinite relaxations of polynomial 0-1 and ±1 programs via scaling
Operations Research Letters
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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 $n\ge 4$ nodes with stability number $\alpha(G)$, n-4 iterations suffice instead of the m (number of edges) iterations required in general and, under some assumption, $n-\alpha(G)-3$ iterations suffice. The exact number of needed iterations is determined for small $n\le 7$ 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 $G^\nabla$ obtained from G by adding a node adjacent to all nodes of G.