Efficient approximation algorithms for semidefinite programs arising from MAX CUT and coloring

Abstract

The best known approximation algorithm for graph MAX CUT, due to Goemans and Williamson, first finds the optimal solution a semidefinite program and then derives a graph cut from that solution. Building on this result, Karger, Motwani, and Sudan gave an approximation algorithm for graph coloring that also involves solving a semidefinite program. Solving these semidefinite programs using known methods (ellipsoid, interiorpoint), though polynomial-time, is quite expensive. We show how they can be approximately solved in Õ(nm) time for graphs with n nodes and m edges.