Augmenting forests to meet odd diameter requirements

  • Authors:

  • Toshimasa Ishii;Shigeyuki Yamamoto;Hiroshi Nagamochi

  • Affiliations:

  • Department of Information and Computer Sciences, Toyohashi University of Technology, Aichi 441-8580, Japan;I FOR COM Co., Ltd. Kanagawa 220-0207, Japan;Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

  • Venue:
  • Discrete Optimization
  • Year:
  • 2006

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Abstract

Given a graph G=(V,E) and an integer D=1, we consider the problem of augmenting G by the smallest number of new edges so that the diameter becomes at most D. It is known that no constant approximation algorithms to this problem with an arbitrary graph G can be obtained unless P=NP. For a forest G and an odd D=3, it was open whether the problem is approximable within a constant factor. In this paper, we give the first constant factor approximation algorithm to the problem with a forest G and an odd D; our algorithm delivers an 8-approximate solution in O(|V|^3) time. We also show that a 4-approximate solution to the problem with a forest G and an odd D can be obtained in linear time if the augmented graph is additionally required to be biconnected.