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.

翻译：调度与近似算法中一个著名的问题是圣诞老人问题。假设圣诞老人拥有一组礼物，他希望将这些礼物分发给一组儿童，使得最不快乐儿童的快乐程度尽可能最大化。其中，儿童$i$对礼物$j$的估值具有$p_{ij} \in \{ 0,p_j\}$的形式。Annamalai等人提出的多项式时间算法基于对Haxell超图匹配论证的改进，达到了12.33倍近似比。本文引入了圣诞老人问题的拟阵版本。我们的算法同样基于Haxell增广树，但通过引入拟阵结构，我们以更简洁的方法解决了更一般的问题。所得结果可作为黑箱工具，为原始圣诞老人问题提供$(6+\varepsilon)$倍近似解。该系数亦可与圣诞老人问题的自然紧凑线性规划松弛解进行比较。