Approximating the minmax rooted-tree cover in a tree

Given an edge-weighted rooted tree T and a positive integer p (⩽n), where n is the number of vertices in T, we cover all vertices in T by a set of p subtrees each of which contains the root r of T. The minmax rooted-tree cover problem asks to find such a set of p subtrees so as to minimize the maximum weight of the subtrees in the set. In this paper, we propose an O(n) time (2+ε)-approximation algorithm to the problem, where ε>0 is a prescribed constant.