Minimum Clique Partition Problem with Constrained Weight for Interval Graphs

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.

The work is fully supported by the National Natural Science Foundation of China [Project No. 10561109, 10271103] and Natural Science Foundation of Yunnan Province [Project No. 2003F0015M]. The partial work of this author was done while visiting Department of Computer Science, City University of Hong Kong.