Boltzmann sampling and optimal exact-size sampling for directed acyclic graphs

TL;DR

This paper introduces two efficient algorithms for uniformly generating directed acyclic graphs, with an asymptotically optimal exact-size sampler that improves theoretical complexity and practical speed over previous methods.

Contribution

The authors extend the Boltzmann model for generating functions and develop new decompositions to create faster, more efficient uniform sampling algorithms for directed acyclic graphs.

Findings

Asymptotically optimal sampler with $rac{n^2}{2} + o(n^2)$ operations

Significant practical speed-up over existing algorithms

Improved theoretical complexity for uniform DAG sampling

Abstract

We propose two efficient algorithms for generating uniform random directed acyclic graphs, including an asymptotically optimal exact-size sampler that performs $\frac{n^2}{2} + o(n^2)$ operations and requests to a random generator. This was achieved by extending the Boltzmann model for graphical generating functions and by using various decompositions of directed acyclic graphs. The presented samplers improve upon the state-of-the-art algorithms in terms of theoretical complexity and offer a significant speed-up in practice.