TY - JOUR

T1 - On listing, sampling, and counting the chordal graphs with edge constraints

AU - Kijima, Shuji

AU - Kiyomi, Masashi

AU - Okamoto, Yoshio

AU - Uno, Takeaki

N1 - Funding Information:
The authors thank the anonymous referee for valuable comments. The authors are supported by Grant-in-Aid for Scientific Research. The third author is also supported by JSPS Global COE Program ‘‘Computationism as a Foundation for the Sciences’’.

PY - 2010/6/6

Y1 - 2010/6/6

N2 - 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 that at least one of the input graphs 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 indicate 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.

AB - 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 that at least one of the input graphs 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 indicate 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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U2 - 10.1016/j.tcs.2010.03.024

DO - 10.1016/j.tcs.2010.03.024

M3 - Article

AN - SCOPUS:77953288766

VL - 411

SP - 2591

EP - 2601

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 26-28

ER -