A projection technique for partitioning the nodes of a graph

Let G = (N, E) be an undirected graph with node set N = { 1 ..... n }, edge set E and edge weights {c(e) : e ~ E}. (If the graph is unweighted, then we assume c(e) = 1 for e E E.) A common problem in circuit board and microchip design, computer program segmentation, floor planning and other layout problems is to partition the node set N into k disjoint subsets Sl ..... Sk of specified sizes ml > m2 > ... > mk, Z k j=lmj = n, so as to minimize the weight of edges connecting nodes in distinct subsets of the partition. We refer to an edge, which connects nodes in distinct subsets of the partition, as being cut by the partition. A recent survey on the graph partitioning problem and further related problems is contained in [21]. There are several possibilities to model graph partition problems. The polyhedral approach relies on linear programming. In this approach, variables are introduced on the edge set E, leading to a linear objective function. The hard part consists in finding linear descriptions of those edge sets which correspond to (feasible) partitions, see e.g. [6]. In the present paper, we do not pursue this approach but use a model based on the node set of the graph. In this case, the description of partitions is straightforward, but the objective function will turn out to be quadratic. The main contribution of this paper will be a theoretically and practically efficient procedure to deal with the quadratic objective function under suitable relaxations of the feasible region.