Counting and Sampling Labeled Chordal Graphs in Polynomial Time

We present the first polynomial-time algorithm to exactly compute the number of labeled chordal graphs on $n$ vertices. Our algorithm solves a more general problem: given $n$ and $\omega$ as input, it computes the number of $\omega$-colorable labeled chordal graphs on $n$ vertices, using $O(n^7)$ arithmetic operations. A standard sampling-to-counting reduction then yields a polynomial-time exact sampler that generates an $\omega$-colorable labeled chordal graph on $n$ vertices uniformly at random. Our counting algorithm improves upon the previous best result by Wormald (1985), which computes the number of labeled chordal graphs on $n$ vertices in time exponential in $n$. An implementation of the polynomial-time counting algorithm gives the number of labeled chordal graphs on up to $30$ vertices in less than three minutes on a standard desktop computer. Previously, the number of labeled chordal graphs was only known for graphs on up to $15$ vertices. In addition, we design two approximation algorithms: (1) an approximate counting algorithm that computes a $(1\pm \epsilon )$-approximation of the number of $n$-vertex labeled chordal graphs, and (2) an approximate sampling algorithm that generates a random labeled chordal graph according to a distribution whose total variation distance from the uniform distribution is at most $\epsilon$. The approximate counting algorithm runs in $O(n^3\log n{\log }^7(1\mathrm{/}\epsilon ))$ time, and the approximate sampling algorithm runs in $O(n^3\log n{\log }^7(1\mathrm{/}\epsilon ))$ expected time.