Minmax Tree Cover in the Euclidean Space

  • Authors:

  • Seigo Karakawa;Ehab Morsy;Hiroshi Nagamochi

  • Affiliations:

  • Department of Applied Mathematics and Physics Graduate School of Informatics, Kyoto University, Kyoto, Japan 606-8501;Department of Applied Mathematics and Physics Graduate School of Informatics, Kyoto University, Kyoto, Japan 606-8501;Department of Applied Mathematics and Physics Graduate School of Informatics, Kyoto University, Kyoto, Japan 606-8501

  • Venue:
  • WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
  • Year:
  • 2009

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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