Analytic Samplers and the Combinatorial Rejection Method

TL;DR

This paper introduces Analytic Samplers, a flexible variant of Boltzmann samplers that tolerates approximation errors, simplifying the calibration process for uniform random sampling of combinatorial objects.

Contribution

It adapts the rejection method to create a new class of samplers that are tolerant of approximation, enabling exact sampling and easier tuning.

Findings

Allows exact sampling with approximate generating function values

Reduces complexity in calibrating samplers for combinatorial classes

Demonstrates effectiveness on simple trees

Abstract

Boltzmann samplers, introduced by Duchon et al. in 2001, make it possible to uniformly draw approximate size objects from any class which can be specified through the symbolic method. This, through by evaluating the associated generating functions to obtain the correct branching probabilities. But these samplers require generating functions, in particular in the neighborhood of their sunglarity, which is a complex problem; they also require picking an appropriate tuning value to best control the size of generated objects. Although Pivoteau~\etal have brought a sweeping question to the first question, with the introduction of their Newton oracle, questions remain. By adapting the rejection method, a classical tool from the random, we show how to obtain a variant of the Boltzmann sampler framework, which is tolerant of approximation, even large ones. Our goal for this is twofold: this allows for exact sampling with approximate values; but this also allows much more flexibility in tuning samplers. For the class of simple trees, we will try to show how this could be used to more easily calibrate samplers.