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Linear-Time Counting Algorithms for Independent Sets in Chordal Graphs

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Graph-Theoretic Concepts in Computer Science (WG 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3787))

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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.

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Author information

Authors and Affiliations

  1. Department of Information and Computer Sciences, Toyohashi University of Technology, Hibarigaoka 1-1, Tempaku, Toyohashi, Aichi, 441-8580, Japan

    Yoshio Okamoto

  2. National Institute of Informatics, Hitotsubashi 2-1-2, Chiyoda-ku, Tokyo, 101-8430, Japan

    Takeaki Uno

  3. School of Information Science, JAIST, Asahidai 1-1, Nomi, Ishikawa, 923-1292, Japan

    Ryuhei Uehara

Authors
  1. Yoshio Okamoto
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  2. Takeaki Uno
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  3. Ryuhei Uehara
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Editor information

Editors and Affiliations

  1. LITA, Université Paul Verlaine-Metz, 57045, Metz Cedex 01, France

    Dieter Kratsch

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© 2005 Springer-Verlag Berlin Heidelberg

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Okamoto, Y., Uno, T., Uehara, R. (2005). Linear-Time Counting Algorithms for Independent Sets in Chordal Graphs. In: Kratsch, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2005. Lecture Notes in Computer Science, vol 3787. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11604686_38

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  • DOI: https://doi.org/10.1007/11604686_38

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-31000-6

  • Online ISBN: 978-3-540-31468-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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