Constructive Heuristics and Lower Bounds for Graph Partitioning Based on a Principal-Components Approximation

Abstract

This paper addresses the problem of partitioning the vertex set of an edge-weighted undirected graph into two parts of specified sizes so as to minimize the sum of the weights on edges joining Vertices in different parts. This problem is NP-hard and has several important applications in which the graph size is typically large and the brute-force approach (of listing all feasible partitions and comparing costs) is computationally prohibitive. In this paper a new class of algorithms is developed on the basis of a transformation of the graph problem to a geometric problem of clustering a set of points in Euclidean space. Instead of searching through all feasible partitions that meet the size specifications, it is shown that the search can be confined to a set of $n^{p( p + 1 )/2} $ partitions, where n is the number of vertices in the graph and p is the rank of the $n \times n$ graph connection matrix. Procedures are developed for constructing all such partitions in $O( n^{p( p + 3 ) /2} )$ time. For matrices...