A Tale of Santa Claus, Hypergraphs and Matroids

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 $i$ has for a present $j$ is of the form $p_{ij} \in \{ 0,p_j\}$. 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 $(6+\varepsilon)$-approximation for Santa Claus. This factor also compares against a natural, compact LP for Santa Claus.