Santa Claus, Machine Scheduling and Bipartite Hypergraphs

Here we discuss two related discrete optimization problems, a prominent problem in scheduling theory, makespan minimization on unrelated parallel machines, and the other a fair allocation problem, the Santa Claus problem. In each case the objective is to make the least well off participant as well off as possible, and in each case we have complexity results that bound how close we may estimate optimal values of worst-case instances in polytime. We explore some of the techniques that have been used in obtaining approximation algorithms or optimal value guarantees for these problems, as well as those involved in getting hardness results, emphasizing the relationships between the problems. A framework for decisional variants of approximation and optimal value estimation for optimization problems is introduced to clarify the discussion. Also discussed are bipartite hypergraphs, which correspond naturally to these problems, including a discussion of Haxell's Theorem for bipartite hypergraphs. Conditions for edge covering in bipartite hypergraphs are introduced and their implications investigated. The conditions are motivated by analogy to Haxell's Theorem and from generalizing conditions that arose from bipartite hypergraphs associated with machine scheduling