Linear-time counting algorithms for independent sets in chordal graphs

Authors:
Yoshio Okamoto;Takeaki Uno;Ryuhei Uehara
Affiliations:
Department of Information and Computer Sciences, Toyohashi University of Technology, Toyohashi, Aichi, Japan;national Institute of Informatics, Tokyo, Japan;School of Information Science, JAIST, Ishikawa, Japan
Venue:
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
Year:
2005

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Abstract

We study some counting and enumeration problems for chordal graphs, especially concerning independent sets. We first provide the following efficient algorithms for a chordal graph: (1) a linear-time algorithm for counting the number of independent sets; (2) a linear-time algorithm for counting the number of maximum independent sets; (3) a polynomial-time algorithm for counting the number of independent sets of a fixed size. With similar ideas, we show that enumeration (namely, listing) of the independent sets, the maximum independent sets, and the independent sets of a fixed size in a chordal graph can be done in constant amortized time per output. On the other hand, we prove that the following problems for a chordal graph are # P-complete: (1) counting the number of maximal independent sets; (2) counting the number of minimum maximal independent sets. With similar ideas, we also show that finding a minimum weighted maximal independent set in a chordal graph is nP-hard, and even hard to approximate.