The Optimal Partitioning of Graphs

Abstract

The problem considered in this paper is that of partitioning a link-weighted graph G into two parts, each of which is constrained in size by the (given) maximum number of vertices that the part can contain. This is a special case of the general partitioning problem of a graph into k parts with size constraints, which appears in a number of very diverse problem areas. The paper presents a tree search method for solving the “partition in two” problem, with the objective of minimizing the sum of the costs of the cut links. The method employs a maximum flow subalgorithm for establishing bounds during the search, and uses the concept of the longest spanning tree of a graph to direct the forward branching steps. The size of the search is reduced further by a test (based on the solution of a special “knapsack” type problem), which is designed to detect potential size-infeasibility early in the search. Computational results are given for approximately 300 graphs of various sizes (up to 40 vertices), and for various densities and types. The computational results show that the proposed algorithm is quite an efficient partitioning procedure for sparse graphs of all types (planar, nonplanar, regular or not), but that its performance deteriorates as graph density is increased.