Thresholds for Extreme Orientability

TL;DR

This paper determines the precise density threshold for the existence of (k-1,1)-orientations in hypergraphs, and provides a linear-time algorithm with explicit failure bounds, advancing load balancing theory.

Contribution

It establishes the density threshold for (k-1,1)-orientations and introduces a linear-time algorithm with explicit failure probability bounds.

Findings

Threshold density below which (k-1,1)-orientations exist with high probability.

A linear-time algorithm for finding such orientations with explicit failure bounds.

Extension of algorithms to all k, beyond previous limits of k<=3.

Abstract

Multiple-choice load balancing has been a topic of intense study since the seminal paper of Azar, Broder, Karlin, and Upfal. Questions in this area can be phrased in terms of orientations of a graph, or more generally a k-uniform random hypergraph. A (d,b)-orientation is an assignment of each edge to d of its vertices, such that no vertex has more than b edges assigned to it. Conditions for the existence of such orientations have been completely documented except for the "extreme" case of (k-1,1)-orientations. We consider this remaining case, and establish: - The density threshold below which an orientation exists with high probability, and above which it does not exist with high probability. - An algorithm for finding an orientation that runs in linear time with high probability, with explicit polynomial bounds on the failure probability. Previously, the only known algorithms for constructing (k-1,1)-orientations worked for k<=3, and were only shown to have expected linear running time.