Load balancing and orientability thresholds for random hypergraphs

Let h > w > 0 be two fixed integers. Let H be a random hypergraph whose hyperedges are uniformly of size h. To 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 H consists of a w-orientation of all hyperedges of H, 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, thereby settling a conjecture made by Karp and Saks. Motivated by a problem of cuckoo hashing, the special hypergraph case with w = k = 1, was solved in three separate preprints dating from October 2009, by Frieze and Mel...

• Type: Conference
• Year: 2010
• Where: STOC
• Authors: Pu Gao, Nicholas C. Wormald
• Keywords: Algorithms, Positive Signs, Random Hypergraph, STOC 2010, Vertices Positive Signs