Approximation Algorithms for Min-Max Tree Partition

We consider the problem of partitioning the node set of a graph into p equal 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 an O(n²) time algorithm for the problem, where n is 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 integer x. It runs in O(2^(p+x)pn²) time, and its error ratio is at most (2 -x/ (x + p - 1))p.