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

MAX CUT is the problem of partitioning the vertices of a graph into two sets max imizing the number of edges joining these sets This problem is NP hard Goemans and Williamson proposed an algorithm that rst uses a semide nite 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 sdp where and sdp is the value of the semide nite program which is an upper bound on opt the number of edges in the maximum cut This manuscript shows the following results The integrality ratio of the semide nite program is The previously known bound on the integrality ratio was roughly In the presence of the so called triangle constraints the integrality ratio is no better than roughly The previously known bound was above There are graphs and optimal embeddings for which the best hyperplane approxi mates opt within a ratio no better than even in the presence of additional valid constraints This strengthens a result of Karlo that applied only to the expected number of edges cut by a random hyperplane The Algorithm of Goemans and Williamson For a graph G V E with jV j n and jEj m MAX CUT is the problem of partitioning V into two sets such that the number of edges connecting the two sets is maximized This problem is NP hard to approximate within ratios better than Partitioning the vertices into two sets at random gives a cut whose expected number of edges is m trivially giving an approximation algorithm with expected ratio at least For many years nothing substantially better was known In a major breakthrough Goemans and Williamson gave an algorithm with approximation ratio of For completeness we review their well known algorithm which we call algorithm GW MAX CUT can be formulated as an integer quadratic program With each vertex i we associate a variable xi f g where and can be viewed as the two sides of the cut With an edge i j E we associate the expression xixj which evaluates to if its endpoints are on the same side of the cut and to if its endpoints are on di erent sides of the cut The integer quadratic program for MAX CUT