Balanced partition of minimum spanning trees

To better handle situations where additional resources are available to carry out a task, many problems from the manufacturing industry involve dividing a task into a number of smaller tasks, while optimizing a specific objective function. In this paper we consider the problem of partitioning a given set S of n points in the plane into k subsets, S-1,...,S-k, such that max(1less than or equal toiless than or equal tok) \MST(S-i)\ is minimized. Variants of this problem arise in applications from the shipbuilding industry. We show that this problem is NP-hard, and we also present an approximation algorithm for the problem, in the case when k is a fixed constant. The approximation algorithm runs in time O(n logn) and produces a partition that is within a factor (4/3 + epsilon) of the optimal if k = 2, and a factor (2 + epsilon) otherwise.