Some approximation algorithms for the clique partition problem in weighted interval graphs

  • Authors:

  • Mingxia Chen;Jianbo Li;Jianping Li;Weidong Li;Lusheng Wang

  • Affiliations:

  • Department of Science and Technology, Yunnan University, Kunming 650091, PR China;School of Management and Economics, Kunming University of Science and Technology, Kunming 650090, PR China;Department of Mathematics, Yunnan University, Kunming 650091, PR China;Department of Mathematics, Yunnan University, Kunming 650091, PR China;Department of Computer Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, PR China

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2007

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Abstract

Interval graphs play important roles in analysis of DNA chains in Benzer [S. Benzer, On the topology of the genetic fine structure, Proceedings of the National Academy of Sciences of the United States of America 45 (1959) 1607-1620], restriction maps of DNA in Waterman and Griggs [M.S. Waterman, J.R. Griggs, Interval graphs and maps of DNA, Bulletin of Mathematical Biology 48 (2) (1986) 189-195] and other related areas. In this paper, we study a new combinatorial optimization problem, named the minimum clique partition problem with constrained bounds, in weighted interval graphs. For a weighted interval graph G and a bound B, partition the weighted intervals of this graph G into the smallest number of cliques, such that each clique, consisting of some intervals whose intersection on a real line is not empty, has its weight not beyond B. We obtain the following results: (1) this problem is NP-hard in a strong sense, and it cannot be approximated within a factor 32-@e in polynomial time for any @e0; (2) we design three approximation algorithms with different constant factors for this problem; (3) for the version where all intervals have the same weights, we design an optimal algorithm to solve the problem in linear time.