Boltzmann Samplers, P\'olya Theory, and Cycle Pointing

TL;DR

This paper presents a unified framework combining Pólya theory and Boltzmann sampling to count and generate unlabeled combinatorial structures efficiently, extending existing methods to a broad class of structures.

Contribution

It introduces a general pointing method for unlabeled structures, extending Pólya theory, and unifies Boltzmann samplers for diverse combinatorial families.

Findings

Provides enumerative formulas for various structures

Develops efficient random samplers for complex structures

Unifies previous unlabeled sampling principles

Abstract

We introduce a general method to count unlabeled combinatorial structures and to efficiently generate them at random. The approach is based on pointing unlabeled structures in an "unbiased" way that a structure of size n gives rise to n pointed structures. We extend Polya theory to the corresponding pointing operator, and present a random sampling framework based on both the principles of Boltzmann sampling and on P\'olya operators. All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results. Our method is illustrated on several examples: in each case, we provide enumerative results and efficient random samplers. The approach applies to unlabeled families of plane and nonplane unrooted trees, and tree-like structures in general, but also to families of graphs (such as cacti graphs and outerplanar graphs) and families of planar maps.