Sink-free orientations: a local sampler with applications

TL;DR

This paper introduces efficient algorithms for approximately counting and sampling sink-free orientations in graphs with minimum degree at least 3, utilizing a local sink popping method within a partial rejection sampling framework.

Contribution

It provides the first near-linear time sampling algorithm and improved approximate counting algorithms for sink-free orientations using a novel local sink popping approach.

Findings

Deterministic approximate counting algorithm with specific runtime.

Near-linear time sampling algorithm for sink-free orientations.

Randomized approximate counting algorithm with quadratic runtime.

Abstract

For sink-free orientations in graphs of minimum degree at least $3$, we show that there is a deterministic approximate counting algorithm that runs in time $O((n^{73}/\varepsilon^{72})\log(n/\varepsilon))$, a near-linear time sampling algorithm, and a randomised approximate counting algorithm that runs in time $O((n/\varepsilon)^2\log(n/\varepsilon))$, where $n$ denotes the number of vertices of the input graph and $0<\varepsilon<1$ is the desired accuracy. All three algorithms are based on a local implementation of the sink popping method (Cohn, Pemantle, and Propp, 2002) under the partial rejection sampling framework (Guo, Jerrum, and Liu, 2019).