A new approach to the orientation of random hypergraphs

TL;DR

This paper determines the threshold for (l,k)-orientability in random h-uniform hypergraphs, extending previous results and applicable to load balancing and cuckoo hashing, using advanced probabilistic and combinatorial methods.

Contribution

It introduces a new threshold analysis for (l,k)-orientability in broad classes of random hypergraphs, expanding beyond uniform hypergraphs.

Findings

Established the threshold for (l,k)-orientability in random hypergraphs.

Extended the analysis to a broader class of hypergraphs beyond uniform cases.

Provided insights applicable to load balancing and cuckoo hashing scenarios.

Abstract

A h-uniform hypergraph H=(V,E) is called (l,k)-orientable if there exists an assignment of each hyperedge e to exactly l of its vertices such that no vertex is assigned more than k hyperedges. Let H_{n,m,h} be a hypergraph, drawn uniformly at random from the set of all h-uniform hypergraphs with n vertices and m edges. In this paper, we determine the threshold of the existence of a (l,k)-orientation of H_{n,m,h} for k>=1 and h>l>=1, extending recent results motivated by applications such as cuckoo hashing or load balancing with guaranteed maximum load. Our proof combines the local weak convergence of sparse graphs and a careful analysis of a Gibbs measure on spanning subgraphs with degree constraints. It allows us to deal with a much broader class than the uniform hypergraphs.