Sampling Unlabeled Chordal Graphs in Expected Polynomial Time
Úrsula Hébert-Johnson, Daniel Lokshtanov
TL;DR
The paper resolves the problem of uniformly sampling unlabeled chordal graphs on $n$ vertices in expected polynomial time by leveraging a Frobenius–Burnside framework and extending Wormald’s unlabeled-graph sampling approach. It builds two main technical pillars: an $ extsf{FPT}$ counting/sampling algorithm for labeled chordal graphs with a fixed automorphism $ ho$, parameterized by the moved-point count $ u$, and a probabilistic bound showing that the probability a random labeled chordal graph has a specified automorphism is at most $1/2^{c\maxig\{ u^2,nig\}}$, which keeps the expected time tractable. The counting is achieved via a refined dynamic-programming framework that reduces to connected components and tracks how moved vertices behave under automorphisms, with running time $O(2^{7 u}n^9)$ for the labeled automorphism problems. The combination of evaporation-sequence structure, PEO properties, and the Burnside-based sampling yields an $O(n^7)$-time expected unlabeled sampler, providing a concrete pathway toward uniform sampling in GI-complete graph classes and highlighting open directions for further generalization and exact/approximate sampling guarantees.
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 $π$, parameterized by the number of moved points of $π$, and (2) a proof that the probability that a random $n$-vertex labeled chordal graph has a given automorphism $π\in S_n$ is at most $1/2^{c\max\{μ^2,n\}}$, where $μ$ is the number of moved points of $π$ 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 $μ$, but the probability of calling this algorithm with a large value of $μ$ is exponentially small.