Minmax Tree Cover in the Euclidean Space

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 α of maxT∈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, and break the bound 2 on α, showing that α < 2 √ 3 − 3/2 ; 1.964 for d ≥ 3 and α < (13 + √ 109)/12 ; 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. Submitted: April 2009 Reviewed: September 2009 Revised: July 2010 Accepted: October 2010 Final: October 2010 Published: July 2011 Article type: Regular paper Communicated by: S. Das and R. Uehara A preliminary version of the paper appeared in Proceedings of the Third Annual Workshop on Algorithms and Computation (WALCOM), Feb. 18-20, 2009, Kolkata, India, LNCS 5431