Minmax tree cover in the euclidean space

Abstract

Let G = (V,E) be an edge-weighted graph, and let w(H) denote the sum of the weights of the edges in a subgraph H of G. Given a positive integer k, the balanced tree partitioning problem requires to cover all vertices in V by a set T of k trees of the graph so that the ratio a of maxTεT w(T) to w(T*)/k is minimized, where T* denotes a minimum spanning tree of G. The problem has been used as a core analysis in designing approximation algorithms for several types of graph partitioning problems over metric spaces, and the performance guarantees depend on the ratio a of the corresponding balanced tree partitioning problems. It is known that the best possible value of a is 2 for the general metric space. In this paper, we study the problem in the d-dimensional Euclidean space Rd, and break the bound 2 on α, showing that α < 23 - 3/2 = 1.964 for d ≥ 3 and a