Approximation Algorithms for Min–Max Tree Partition

We consider the problem of partitioning the node set of a graph intopequal sized subsets. The objective is to minimize the maximum length, over these subsets, of a minimum spanning tree. We show that no polynomial algorithm with bounded error ratio can be given for the problem unless P = NP. We present anO(n²) time algorithm for the problem, wherenis the number of nodes in the graph. Assuming that the edge lengths satisfy the triangle inequality, its error ratio is at most 2p − 1. We also present an improved algorithm that obtains as an input a positive integerx. It runs inO(2^{(p + x)p}n²) time, and its error ratio is at most (2 − x/(x + p − 1))p.