Tighter linear and semidefinite relaxations for max-cut based on the LovászSchrijver lift-and-project procedure
| Venue: | SIAM Journal on Optimization |
| Citations: | 7 - 4 self |
BibTeX
@ARTICLE{Laurent_tighterlinear,
author = {Monique Laurent},
title = {Tighter linear and semidefinite relaxations for max-cut based on the LovászSchrijver lift-and-project procedure},
journal = {SIAM Journal on Optimization},
year = {},
pages = {345--375}
}
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Abstract
Abstract. 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.
