Sampling Unlabeled Chordal Graphs in Expected Polynomial Time

TL;DR

This paper presents an efficient algorithm for uniformly sampling unlabeled chordal graphs, along with new results on counting and automorphism probabilities of labeled chordal graphs, advancing graph sampling and automorphism analysis.

Contribution

It introduces an expected polynomial time algorithm for sampling unlabeled chordal graphs and develops an FPT algorithm for counting and sampling labeled chordal graphs with a specific automorphism.

Findings

Efficient expected polynomial time sampling algorithm for unlabeled chordal graphs.

An FPT algorithm for counting and sampling labeled chordal graphs with a given automorphism.

Probability bound on automorphisms in random labeled chordal graphs.

Abstract

We design an algorithm that generates an $n$-vertex unlabeled chordal graph uniformly at random in expected polynomial time. Along the way, we develop the following two results: (1) an $\mathsf{FPT}$ algorithm for counting and sampling labeled chordal graphs with a given automorphism $\pi$, parameterized by the number of moved points of $\pi$, and (2) a proof that the probability that a random $n$-vertex labeled chordal graph has a given automorphism $\pi\in S_n$ is at most $1/2^{c\max\{\mu^2,n\}}$, where $\mu$ is the number of moved points of $\pi$ and $c$ is a constant. Our algorithm for sampling unlabeled chordal graphs calls the aforementioned $\mathsf{FPT}$ algorithm as a black box with potentially large values of the parameter $\mu$, but the probability of calling this algorithm with a large value of $\mu$ is exponentially small.