On the integrality ratio of semidefinite relaxations of MAX CUT

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 \cdot sdp, where α \simeq 0.87856 and sdp is the value of the semidefinite program. This manuscript shows the following results: 1. The integrality ratio of the semidefinite program is α. The previously known bound on theintegrality 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.