Sink-free orientations: a local sampler with applications
Konrad Anand, Graham Freifeld, Heng Guo, Chunyang Wang, Jiaheng Wang
TL;DR
This work tackles counting and sampling sink-free orientations (SFOs) in graphs by designing a local sampler based on partial rejection sampling (PRS) and the sink-popping paradigm. The authors develop a generalized local sampler that operates under a substructure S and establish truncation-based efficiency, enabling a deterministic $\varepsilon$-approximation for $|\Omega_V|$ in time $O((n^{73}/\varepsilon^{72})\log(n/\varepsilon))$, a near-linear-time sampler, and a randomized $\varepsilon$-approximation in time $O((n/\varepsilon)^2\log(n/\varepsilon))$ for minimum degree $3$ graphs. They further prove a marginal lower bound $\mu_S(v\text{ not a sink})>\tfrac{1}{2}$ via a Symmetric Shearer bound and connect the SFO count to the independence polynomial evaluated at negative weights, discussing applicable FPTAS approaches within Shearer’s region. The results advance fast approximate counting and sampling for an extremal local-lemma instance, with implications for derandomisation and potential extensions to broader extremal problems such as all-terminal reliability. The work combines local sampling, martingale analyses, and combinatorial polynomial techniques to achieve its results. $\,$
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).