A new approach to the orientation of random hypergraphs

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.

💡 Research Summary

The paper studies the existence of an ((l,k))-orientation in random (h)-uniform hypergraphs. An ((l,k))-orientation assigns each hyperedge to exactly (l) of its incident vertices while ensuring that no vertex receives more than (k) assignments. This problem models practical systems such as cuckoo hashing, load‑balancing with bounded load, and distributed storage where each item must be placed in a limited number of locations and each location can store only a bounded number of items.

The authors consider the random hypergraph model (\mathcal{H}_{n,m,h}), where (n) vertices and (m) hyperedges are chosen uniformly among all (h)-uniform hypergraphs. The central question is: for fixed integers (h>l\ge 1) and (k\ge 1), what is the critical edge‑density (c=m/n) at which a ((l,k))-orientation appears with high probability (whp) as (n\to\infty)? Prior work answered this only for special cases (typically (k=1) or (h=l+1)). This work extends the analysis to the full range (k\ge 1), (h>l\ge 1).

Methodological Innovation

The proof combines two powerful probabilistic tools:

1. Local Weak Convergence – The local neighbourhood of a uniformly random vertex in a sparse hypergraph converges in distribution to a Galton‑Watson branching process (a Poisson‑(h) tree). This reduces the global orientation problem to a recursive problem on an infinite random tree.

2. Gibbs Measure on Constrained Subgraphs – The authors define a Gibbs distribution over spanning sub‑hypergraphs where each vertex’s degree is limited to (k) and each hyperedge is either “selected” (contributing to the orientation) or not. The energy function penalises violations of the degree constraints. By letting the inverse temperature (\beta\to\infty), the Gibbs measure concentrates on configurations that are genuine ((l,k))-orientations.

On the limiting tree, the Gibbs measure induces a message‑passing (belief‑propagation) recursion. For a vertex of degree (d), the incoming messages (\mu_1,\dots,\mu_{d-1}\in