We consider the problem of fairly allocating a sequence of indivisible items that arrive online in an arbitrary order to a group of n agents with additive normalized valuation functions. We consider both the allocation of goods and chores and propose algorithms for approximating maximin share (MMS) allocations. When agents have identical valuation functions the problem coincides with the semi-online machine covering problem (when items are goods) and load balancing problem (when items are chores), for both of which optimal competitive ratios have been achieved. In this paper, we consider the case when agents have general additive valuation functions. For the allocation of goods, we show that no competitive algorithm exists even when there are only three agents and propose an optimal 0.5-competitive algorithm for the case of two agents. For the allocation of chores, we propose a (2-1/n)-competitive algorithm for n>=3 agents and a square root of 2 (approximately 1.414)-competitive algorithm for two agents. Additionally, we show that no algorithm can do better than 15/11 (approximately 1.364)-competitive for two agents.

翻译：我们考虑将按任意顺序在线到达的一系列不可分物品公平分配给一组具有可加归一化估值函数的n个智能体的问题。我们同时考察了物品与杂务的分配，并提出了近似最大化最小份额（MMS）分配的算法。当智能体具有相同的估值函数时，该问题分别转化为半在线机器覆盖问题（当物品为正向商品时）和负载均衡问题（当物品为杂务时），两者均已实现最优竞争比。本文中，我们考虑智能体具有一般可加估值函数的情形。对于正向商品的分配，我们证明即使只有三个智能体也不存在竞争性算法，并为两个智能体的情形提出一个最优的0.5-竞争比算法。对于杂务的分配，我们为n≥3个智能体提出一个(2-1/n)-竞争比算法，为两个智能体提出一个根号2（约1.414）-竞争比算法。此外，我们证明对于两个智能体的情形，任何算法的竞争比都不得优于15/11（约1.364）。