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On Listing, Sampling, and Counting the Chordal Graphs with Edge Constraints

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Computing and Combinatorics (COCOON 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5092))

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Abstract

We discuss the problems to list, sample, and count the chordal graphs with edge constraints. The objects we look at are chordal graphs sandwiched by a given pair of graphs where we assume at least one of the input pair is chordal. The setting is a natural generalization of chordal completions and deletions. For the listing problem, we give an efficient algorithm running in polynomial time per output with polynomial space. As for the sampling problem, we give two clues that seem to imply that a random sampling is not easy. The first clue is that we show #P-completeness results for counting problems. The second clue is that we give an instance for which a natural Markov chain suffers from an exponential mixing time. These results provide a unified viewpoint from algorithms theory to problems arising from various areas such as statistics, data mining, and numerical computation.

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Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Shuji Kijima

  2. School of Information Science, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa, 923-1292, Japan

    Masashi Kiyomi

  3. Graduate School of Information Science and Engineering, Tokyo Institute of Technology, 2-12-1-W8-88, Ookayama, Meguro-ku, Tokyo, 152-8552, Japan

    Yoshio Okamoto

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

    Takeaki Uno

Authors
  1. Shuji Kijima
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  2. Masashi Kiyomi
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  3. Yoshio Okamoto
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  4. Takeaki Uno
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Xiaodong Hu Jie Wang

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Kijima, S., Kiyomi, M., Okamoto, Y., Uno, T. (2008). On Listing, Sampling, and Counting the Chordal Graphs with Edge Constraints. In: Hu, X., Wang, J. (eds) Computing and Combinatorics. COCOON 2008. Lecture Notes in Computer Science, vol 5092. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69733-6_45

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  • DOI: https://doi.org/10.1007/978-3-540-69733-6_45

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-69732-9

  • Online ISBN: 978-3-540-69733-6

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