The Santa Claus problem is a fundamental problem in fair division: the goal is to partition a set of heterogeneous items among heterogeneous agents so as to maximize the minimum value of items received by any agent. In this paper, we study the online version of this problem where the items are not known in advance and have to be assigned to agents as they arrive over time. If the arrival order of items is arbitrary, then no good assignment rule exists in the worst case. However, we show that, if the arrival order is random, then for $n$ agents and any $\varepsilon > 0$, we can obtain a competitive ratio of $1-\varepsilon$ when the optimal assignment gives value at least $\Omega(\log n / \varepsilon^2)$ to every agent (assuming each item has at most unit value). We also show that this result is almost tight: namely, if the optimal solution has value at most $C \ln n / \varepsilon$ for some constant $C$, then there is no $(1-\varepsilon)$-competitive algorithm even for random arrival order.

翻译：圣诞老人问题是公平分配中的一个基础性问题：目标是将一组异质性物品分配给异质性智能体，使得任何智能体获得的物品最小价值最大化。本文研究了该问题的在线版本，其中物品并非事先已知，而是随时间到达需要分配给智能体。如果物品到达顺序是任意的，则最坏情况下不存在好的分配规则。然而，我们证明若到达顺序是随机的，则对于$n$个智能体和任意$\varepsilon > 0$，当最优分配给予每个智能体价值至少为$\Omega(\log n / \varepsilon^2)$时（假设每个物品价值至多为单位1），我们可以获得$1-\varepsilon$的竞争比。我们还证明该结果近乎紧致：即若最优解的价值至多为某个常数$C$的$C \ln n / \varepsilon$，则即使对于随机到达顺序，也不存在$(1-\varepsilon)$-竞争算法。