Counting and Sampling Labeled Chordal Graphs in Polynomial Time

TL;DR

The first polynomial-time algorithm to exactly compute the number of labeled chordal graphs on $n$ vertices is presented, and two approximation algorithms are designed that generate a random labeled chordal graph according to a distribution whose total variation distance from the uniform distribution is at most $\varepsilon".

Abstract

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 $ω$ as input, it computes the number of $ω$-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 $ω$-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 $(1\pm\varepsilon)$-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 $\varepsilon$. The approximate counting algorithm runs in $O(n^3\log{n}\log^7(1/\varepsilon))$ time, and the approximate sampling algorithm runs in $O(n^3\log{n}\log^7(1/\varepsilon))$ expected time.

Figures (2)

• Figure 1: The counter functions. Here $L = L_G(X)$ and $Z=[z]$. An arrow from one function to another, say from $\tilde{g}$ to $g$, indicates that the definition of $\tilde{g}$ is the same as that of $g$, except where indicated otherwise. The drawing of $\tilde{f}_p$ represents $\tilde{f}_p(t,x,l,k)$. The function $\tilde{f}_p(t,x,l,k,z)$ is similar but also keeps track of the argument $z$.
• Figure 2: The control flow of the counting algorithm. An arrow from $f$ to $g$ indicates that the recursive formula for $f$ depends on $g$. The arrow is red (and dashed) if $t$ decreases by 1, blue (and dashed) if $t$ decreases by 2, and black if $t$ does not change. The black self-loops do not cause an infinite loop because in those recursive calls, the number of vertices decreases.

Theorems & Definitions (103)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 3.1
• Theorem 3.2
• Definition 3.3
• Lemma 3.3
• Lemma 3.3
• Lemma 3.3