On the optimality of the random hyperplane rounding technique for MAX CUT

Abstract

Abstract MAX CUT is the problem of partitioning the vertices of a graph into two sets, maximizing the number of edges joining these sets. This problem is NP‐hard. Goemans and Williamson proposed an algorithm that first uses a semidefinite programming relaxation of MAX CUT to embed the vertices of the graph on the surface of an n ‐dimensional sphere, and then uses a random hyperplane to cut the sphere in two, giving a cut of the graph. They show that the expected number of edges in the random cut is at least α · sdp, where α ≃ 0.87856 and sdp is the value of the semidefinite program, which is an upper bound on opt, the number of edges in the maximum cut. This manuscript shows the following results: (1) The integrality ratio of the semidefinite program is α. The previously known bound on the integrality ratio was roughly 0.8845. (2) In the presence of the so‐called “triangle constraints,” the integrality ratio is no better than roughly 0.891. The previously known bound was above 0.95. (3) There are graphs and optimal embeddings for which the best hyperplane approximates opt within a ratio no better than α, even in the presence of additional valid constraints. This strengthens a result of Karloff that applied only to the expected number of edges cut by a random hyperplane. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 403–440, 2002