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 edge constraints are given as a pair of graphs one of which contains the other and one of which is chordal, and the objects we look at are the chordal graphs contained in one and containing the other. This setting is a natural generalization of chordal completions and deletions. For the listing problem, we give an efficient algorithm running in amortized polynomial time per output with polynomial space. For the sampling problem, we give an instance for which a natural Markov chain suffers from an exponential mixing time. For the counting problem, we show some #P-completeness results. These results provide a unified viewpoint from algorithms theory to problems arising from various areas such as statistics, data mining, and numerical computation. key words: graph sandwich, chordal completion/deletion, enumeration, Markov chain Monte Carlo, #P-completeness.