Sampling Arborescences in Parallel

TL;DR

This paper demonstrates that sampling arborescences in directed graphs can be efficiently performed in parallel, advancing the understanding of parallel algorithms for combinatorial structures.

Contribution

It introduces a parallel algorithm in RNC for sampling arborescences, addressing the sequential bottleneck in classical sampling methods.

Findings

Sampling arborescences is in RNC.

Parallel algorithms for combinatorial structures are feasible.

Addresses the sequential nature of classical sampling methods.

Abstract

We study the problem of sampling a uniformly random directed rooted spanning tree, also known as an arborescence, from a possibly weighted directed graph. Classically, this problem has long been known to be polynomial-time solvable; the exact number of arborescences can be computed by a determinant [Tut48], and sampling can be reduced to counting [JVV86, JS96]. However, the classic reduction from sampling to counting seems to be inherently sequential. This raises the question of designing efficient parallel algorithms for sampling. We show that sampling arborescences can be done in RNC. For several well-studied combinatorial structures, counting can be reduced to the computation of a determinant, which is known to be in NC [Csa75]. These include arborescences, planar graph perfect matchings, Eulerian tours in digraphs, and determinantal point processes. However, not much is known about efficient parallel sampling of these structures. Our work is a step towards resolving this mystery.