A Tale of Santa Claus, Hypergraphs and Matroids
Sami Davies, Thomas Rothvoß, Yihao Zhang · Society for Industrial and Applied Mathematics eBooks · 2019
A well-known problem in scheduling and approximation algorithms is the Santa Claus problem. Suppose that Santa Claus has a set of gifts, and he wants to distribute them among a set of children so that the least happy child is made as happy as possible. Here, the value that a child i has for a present j is of the form pij ϵ {0, pj}. A polynomial time algorithm by Annamalai et al. gives a 12.33-approximation and is based on a modification of Haxell's hypergraph matching argument. In this paper, we introduce a matroid version of the Santa Claus problem. Our algorithm is also based on Haxell's augmenting tree, but with the introduction of the matroid structure we solve a more general problem with cleaner methods. Our result can then be used as a blackbox to obtain a (4 + ϵ)-approximation for Santa Claus. This factor also compares against a natural, compact LP for Santa Claus.