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Linear-Time Counting Algorithms for Independent Sets in Chordal Graphs

  • Yoshio Okamoto
  • Takeaki Uno
  • Ryuhei Uehara
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3787)

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.

Keywords

Chordal graph counting enumeration independent set NP-completeness # P-completeness polynomial time algorithm 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Yoshio Okamoto
    • 1
  • Takeaki Uno
    • 2
  • Ryuhei Uehara
    • 3
  1. 1.Department of Information and Computer SciencesToyohashi University of TechnologyToyohashi, AichiJapan
  2. 2.National Institute of InformaticsTokyoJapan
  3. 3.School of Information ScienceJAISTIshikawaJapan

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