Random generation of combinatorial structures: Boltzmann samplers and beyond

TL;DR

This paper discusses the Boltzmann model for efficiently generating random decomposable combinatorial structures, highlighting its theoretical foundations and recent advancements in the field.

Contribution

It provides a comprehensive overview of the Boltzmann sampling technique and introduces newer developments beyond the classical methods.

Findings

Efficient algorithms for random generation of combinatorial objects.

Relaxation of uniformity requirement while maintaining size-based uniformity.

Integration of generating functions as a key tool in sampling methods.

Abstract

The Boltzmann model for the random generation of "decomposable" combinatorial structures is a set of techniques that allows for efficient random sampling algorithms for a large class of families of discrete objects. The usual requirement of sampling uniformly from the set of objects of a given size is somehow relaxed, though uniformity among objects of each size is still ensured. Generating functions, rather than the enumeration sequences they are based on, are the crucial ingredient. We give a brief description of the general theory, as well as a number of newer developments.