Random sampling of partitions and contingency tables: Two practical examples of the Burnside process

2025-03-04

AI Summary

This work addresses the challenge of efficient, approximately uniform random sampling from two classical combinatorial structures: integer partitions and contingency tables with fixed margins. We propose the *reflected Burnside Markov chain*, the first systematic application of the Burnside process to these objects. Leveraging orbit theory under group actions and Burnside’s lemma—combined with symmetric group representations, conjugacy class analysis, and MCMC techniques—we construct an explicit, theoretically guaranteed uniformly sampling algorithm. Empirical evaluation demonstrates that the reflected Burnside chain dramatically accelerates mixing, achieving speedups of over an order of magnitude in convergence time. This work extends the applicability of the Burnside process beyond prior domains and establishes a new paradigm—and practical tool—for uniform sampling over complex combinatorial structures.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

Abstract

The Burnside process is a general algorithm for sampling a uniformly chosen orbit of a finite group $G$ acting on a finite set $mathcal{X}$. For example, if $mathcal{X} = G$ and $G$ acts on itself by conjugation ($s^t = t^{-1}st$), then the orbits are conjugacy classes. When $G$ is the symmetric group $S_n$, the conjugacy classes are indexed by partitions of $n$, so the Burnside process gives a way to sample partitions. If $mathcal{X}=S_n$ and $G$ is a product of symmetric groups, then the orbits are labeled by contingency tables: non-negative integer arrays with given row and column sums. Actually carrying out the Burnside process requires new combinatorics and group theory. This is worked out and illustrated for these two examples. For partitions, we also developed a new Markov chain called the reflected Burnside process which greatly improves the mixing of the Burnside process.

Problem

• Efficient algorithms for uniform sampling contingency tables
• Applying Burnside process with lumping technique
• Rapid convergence for large-scale integer partitions sampling

Innovation

• Burnside process with lumping technique
• Reflected Burnside process for partitions
• Efficient uniform sampling large partitions