Orientability of Random Hypergraphs and the Power of Multiple Choices

Abstract

A hypergraph H = (V, E) is called s-orientable, if there is an assignment of each edge e ∈ E to one of its vertices v ∈ e such that no vertex is assigned more than s edges. Let H _n,m,k be a hypergraph, drawn uniformly at random from the set of all k-uniform hypergraphs with n vertices and m edges. In this paper we establish the threshold for the 1-orientability of H _n,m,k for all k ≥ 3, i.e., we determine a critical quantity $c_k^*$ such that with probability 1 − o(1) the graph H _n,cn,k has a 1-orientation if $c < c_k^*$, but fails doing so if $c > c_k^*$.

We present two applications of this result that involve the paradigm of multiple choices. First, we show how it implies sharp load thresholds for cuckoo hash tables, where each element chooses k out of n locations. Particularly, for each k ≥ 3 we prove that with probability 1 − o(1) the maximum number of elements that can be hashed is $(1 - o(1))c_k^* n$, and more items prevent the successful allocation. Second, we study random graph processes, where in each step we have the choice among any edge connecting k random vertices. Here we show the existence of a phase transition for avoiding a giant connected component, and quantify precisely the dependence on k.