Rank-Two Relaxation Heuristics for Max-Cut and Other Binary Quadratic Programs

Samuel Burer, Renato Monteiro and Yin Zhang

Semidefinite relaxation for certain discrete optimization problems involves replacing a vector-valued variable by a matrix-valued one, producing a convex program while increasing the number of variables by an order of magnitude. As useful as it is in theory, this approach encounters difficulty in practice as problem size increases. In this paper, we propose a rank-two relaxation approach and construct continuous optimization heuristics applicable to some binary quadratic programs, including primarily the Max-Cut problem but also others such as the Max-Bisection problem. A computer code based on our rank-two relaxation heuristics is compared with two state-of-the-art semidefinite programming codes. We will report some rather intriguing computational results on a large set of test problems and discuss their ramifications.

Technical Report TR00-33 Department of Computational and Applied Mathematics Rice University, Houston, Texas 77005

Contact: zhang@caam.rice.edu