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 $\mathcal{T}$ of k trees of the graph so that the ratio α of $\max_{T\in \mathcal{T}}w(T)$ 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 α of the corresponding balanced tree partitioning problems. It is known that the best possible value of α is 2 for the general metric space. In this paper, we study the problem in the d-dimensional Euclidean space ℝ^d, and break the bound 2 on α, showing that $\alpha <2\sqrt{3}-3/2 \fallingdotseq 1.964$ for d ≥ 3 and $\alpha <(13 + \sqrt{109})/12 \fallingdotseq 1.953$ for d = 2. These new results enable us to directly improve the performance guarantees of several existing approximation algorithms for graph partitioning problems if the metric space is an Euclidean space.