Orientability thresholds for random hypergraphs

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.

💡 Research Summary

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The paper studies the existence of a ((w,k))-orientation in a random (h)-uniform hypergraph (\mathcal H(n,m)). An orientation assigns exactly (w) positive signs to each hyperedge (the remaining (h-w) vertices receive negative signs). A ((w,k))-orientation is a collection of such assignments in which every vertex receives at most (k) positive signs from its incident hyperedges. This combinatorial problem is equivalent to an offline load‑balancing model where each job (hyperedge) is replicated on exactly (w) machines and each machine can handle at most (k) replicas simultaneously.

The authors introduce the notion of a ((w,k))-core, obtained by repeatedly deleting vertices that already have at most (k) positive signs and hyperedges that cannot provide the required (w) positive signs. The core is the sub‑hypergraph that survives this “peeling” process. The existence of a ((w,k))-orientation is shown to be equivalent to the emptiness of the core: if the core is empty, a feasible orientation can be constructed; if the core is non‑empty, no orientation satisfies the constraints.

To analyse the random process, the paper employs the configuration model for hypergraphs, which preserves the degree distribution while allowing independent random matchings of half‑edges. The peeling process is then modeled as a continuous‑time dynamical system. Using the differential‑equation method, the authors derive a system of ordinary differential equations describing the evolution of the fraction of remaining vertices (x(t)) and hyperedges (y(t)) as a function of the removal time (t). The key equation for the vertex fraction is

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