A COPOSITIVE PROGRAMMING APPROACH TO GRAPH PARTITIONING.

We consider 3-partitioning the vertices of a graph into sets S1, S2, and S3 of specified cardinalities, such that the total weight of all edges joining S1 and S2 is minimized. This problem is closely related to several NP-hard problems like determining the bandwidth or finding a vertex separator in a graph. We show that this problem can be formulated as a linear program over the cone of completely positive matrices, leading in a natural way to semidefinite relaxations of the problem. We show in particular that the spectral relaxation introduced by Helmberg et al. (1995) can equivalently be formulated as a semidefinite program. Finally we propose a tightened version of this semidefinite program and show on some small instances that this new bound is a significant improvement over the spectral bound. [ABSTRACT FROM AUTHOR]