Minimum clique partition problem with constrained weight for interval graphs

Authors:
Jianbo Li;Mingxia Chen;Jianping Li;Weidong Li
Affiliations:
School of Management and Economics, Kunming University of Science and Technology, Kunming, P.R. China;Department of Science and Technology, Yunnan University, Kunming, P.R. China;Department of Mathematics, Yunnan University, Kunming, P.R. China;Department of Mathematics, Yunnan University, Kunming, P.R. China
Venue:
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
Year:
2006

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Abstract

Interval graphs play important roles in analysis of DNA chains in Benzer [1], restriction maps of DNA in Waterman and Griggs [11] and other related areas. In this paper, we study a new combinatorial optimization problem, named as the minimum clique partition problem with constrained weight, for 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, where 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 the strong sense, and it cannot be approximated within a ratio $\frac{3}{2}-\varepsilon$ in polynomial-time for any ε 0; (2) We design some approximation algorithms with different constant ratios to this problem; (3) For the case where all intervals have the same weight, we also design an optimal algorithm to solve the problem in linear time.