A Tale of Santa Claus, Hypergraphs and Matroids

2018-07-19

ACM-SIAM Symposium on Discrete Algorithms

AI Summary

This paper studies the Santa Claus problem with matroid constraints: allocate indivisible items (gifts) to children to maximize the minimum utility among children, where child $i$’s value for item $j$ is either $0$ or $p_j$. It introduces matroid constraints—modeling general resource allocation restrictions—for the first time in this context. Methodologically, it simplifies and generalizes Haxell’s augmenting tree technique, yielding a unified and concise approximation framework that integrates matroid theory, hypergraph matching, linear programming relaxation, and rounding. The algorithm achieves a $(4+varepsilon)$-approximation ratio, substantially improving upon the previous best $12.33$-approximation. Moreover, it serves as a black-box improvement for the LP relaxation bound of the classical Santa Claus problem. This work establishes a new paradigm for fair allocation under combinatorial constraints, bridging fairness, discrete optimization, and structural constraint modeling.

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 $(4+varepsilon)$-approximation for Santa Claus. This factor also compares against a natural, compact LP for Santa Claus.

Problem

• Generalizing Santa Claus problem with matroid structure
• Improving approximation algorithm for gift distribution
• Providing cleaner methods for scheduling optimization

Innovation

• Matroid structure for Santa Claus problem
• Haxell's hypergraph matching modification
• Blackbox for 6+ε approximation algorithm