The random graph threshold for k-orientiability and a fast algorithm for optimal multiple-choice allocation

We investigate a linear time greedy algorithm for the following load balancing problem: Assign m balls to n bins such that the maximum occupancy is minimized. Each ball can be placed into one of two randomly choosen bins. This problem is closely related to the problem of orienting the edges of an undirected graph to obtain a directed graph with minimum in-degree. Using differential equation methods, we derive thresholds for the solution quality achieved by our algorithm. Since these thresholds coincide with lower bounds for the achievable solution quality, this proves the optimality of our algorithm (as n → ∞, in a probabilistic sense) and establishes the thresholds for k-orientability of random graphs. This proves an assertion of Karp and Saks.