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 Ž 2 . error ratio can be given for the problem unless P s NP. We present an O n 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 2 p y 1. We also present an improved algorithm that obtains as an input a positive Ž Ž pqx . p 2 . Ž integer x. It runs in O 2 n time, and its error ratio is at most 2 y xr Ž .. x q p y 1 p. Q 1997 Academic Press