Orientability thresholds for random hypergraphs

TL;DR

This paper determines the threshold for the existence of specific orientations in random hypergraphs, linking combinatorial properties to load balancing problems, especially for large enough k.

Contribution

It extends the understanding of hypergraph orientations by establishing thresholds for existence, generalizing previous graph results to hypergraphs.

Findings

Thresholds for (w,k)-orientations in random hypergraphs are identified.

The work connects hypergraph orientations to off-line load balancing.

Results generalize known graph case to hypergraphs.

Abstract

Let $h>w>0$ be two fixed integers. Let $\orH$ be a random hypergraph whose hyperedges are all of cardinality $h$. To {\em $w$-orient} a hyperedge, we assign exactly $w$ of its vertices positive signs with respect to the hyperedge, and the rest negative. A $(w,k)$-orientation of $\orH$ consists of a $w$-orientation of all hyperedges of $\orH$, such that each vertex receives at most $k$ positive signs from its incident hyperedges. When $k$ is large enough, we determine the threshold of the existence of a $(w,k)$-orientation of a random hypergraph. The $(w,k)$-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The graph case, when $h=2$ and $w=1$, was solved recently by Cain, Sanders and Wormald and independently by Fernholz and Ramachandran, which settled a conjecture of Karp and Saks.