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

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.