Polynomial Time 2-Approximation Algorithms for the Minmax Subtree Cover Problem

Abstract

Let T be a tree such that edges are weighted by nonnegative reals, and p be a positive integer. The minmax subtree cover problem asks to find a set of p subtrees such that the union of the subtrees covers all vertices in T, where the objective is to minimize the maximum weight of the subtrees. Given a root r in T, the minmax rooted-subtree cover problem asks to find a set of p subtrees such that each subtree contains the root r and the union of the subtrees covers all vertices in T, where the objective is to minimize the maximum weight of the subtrees. In this paper, we propose an O(p ² n) time $(2 - \frac{2}{p+1})$-approximation algorithm to the first problem, and an $O(n{\rm log log_{1+\frac{\epsilon}{2}}3})$ time (2+ε)-approximation algorithm to the second problem, where ε> 0 is a prescribed constant.