Sampling Unlabeled Chordal Graphs in Expected Polynomial Time

2025-01-09

AI Summary

This paper addresses the problem of efficient uniform random sampling of unlabeled chordal graphs. For $n$-vertex unlabeled chordal graphs, we present the first uniform sampling algorithm with expected polynomial runtime ($O(n^c)$). Our method introduces fixed-parameter tractable (FPT) counting and sampling subroutines based on the number of moved points under graph automorphisms, integrated within a group-action framework grounded in Burnside’s lemma. By deriving an exponential upper bound on the probability that a random automorphism fixes a given chordal graph, we tightly control sampling bias induced by isomorphism classes. We rigorously prove that the total variation distance between the output distribution and the uniform distribution over unlabeled chordal graphs is negligible. Empirical evaluation confirms sustained efficiency and stability as $n$ increases. To our knowledge, this is the first polynomial-time uniform sampler for unlabeled chordal graphs with both theoretical guarantees and practical viability.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $piin S_n$ is at most $1/2^{cmax{mu^2,n}}$, 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.

Problem

• Unsupervised Learning
• Graph Identification
• String Graphs

Innovation

• Polynomial-time Algorithm
• Random Generation
• Unlabeled String Graphs