An Efficient Semidefinite Programming Relaxation for the Graph Partition Problem

We derive a new semidefinite programming relaxation for the general graph partition problem (GPP). Our relaxation is based on matrix lifting with matrix variable having order equal to the number of vertices of the graph. We show that this relaxation is equivalent to the Frieze-Jerrum relaxation [A. Frieze and M. Jerrum. Improved approximation algorithms for max k-cut and max bisection. Algorithmica, 18(1):67–81, 1997] for the maximum k-cut problem with an additional constraint that involves the restrictions on the subset sizes. Since the new relaxation does not depend on the number of subsets k into which the graph should be partitioned we are able to compute bounds for large k. We compare theoretically and numerically the new relaxation with other SDP relaxations for the GPP. The results show that our relaxation provides competitive bounds and is solved significantly faster than any other known SDP bound for the general GPP.