Bipartitioning into Overlapping Sets

We consider the problem of partitioning a graph G = (V; E) into two sets V 1 and V 2 such that jV 1 \ V 2 j is no more than an integer d, and such that P u2V 1 ?V 2 ;v2V 2 ?V 1 (u; v) is minimized. We show that this problem is NP-hard in general. It remains NP-hard if jV 1 j = jV 2 j, or if we insist that, for any bipartition, v 1 2 V 1 and v 2 2 V 2 for two speciic nodes v 1 ; v 2 2 V. The problem variation in which either jV 1 j = jV 2 j or jV 1 j = k, k 2 Z + , nds important applications in VLSI layout and hypertext partitioning. We examine the latter problems on special cases of graphs which have been examined in the literature for similar partitioning problems. We present polynomial time algorithms for the special cases of (a) series-parallel graphs, and (b) solid grids.