A Tale of Santa Claus, Hypergraphs and Matroids
Sami Davies, Thomas Rothvoss, Yihao Zhang

TL;DR
This paper introduces a matroid-based generalization of the Santa Claus problem, providing a cleaner algorithmic approach and achieving a better approximation ratio of (6+ε) compared to previous methods.
Contribution
It develops a matroid version of the Santa Claus problem and offers a new algorithm that improves approximation guarantees using Haxell's augmenting tree.
Findings
Achieves a (6+ε)-approximation for Santa Claus.
Introduces a matroid framework for the problem.
Provides a cleaner, more general algorithmic approach.
Abstract
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 has for a present is of the form . A polynomial time algorithm by Annamalai et al. gives a -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 -approximation for Santa Claus. This factor also compares against a natural, compact LP for…
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Taxonomy
TopicsOptimization and Search Problems · Advanced Graph Theory Research · Complexity and Algorithms in Graphs
