We study the fair allocation of indivisible goods among agents with identical, additive valuations but individual budget constraints. Here, the indivisible goods--each with a specific size and value--need to be allocated such that the bundle assigned to each agent is of total size at most the agent's budget. Since envy-free allocations do not necessarily exist in the indivisible goods context, compelling relaxations--in particular, the notion of envy-freeness up to $k$ goods (EFk)--have received significant attention in recent years. In an EFk allocation, each agent prefers its own bundle over that of any other agent, up to the removal of $k$ goods, and the agents have similarly bounded envy against the charity (which corresponds to the set of all unallocated goods). Recently, Wu et al. (2021) showed that an allocation that satisfies the budget constraints and maximizes the Nash social welfare is $1/4$-approximately EF1. However, the computation (or even existence) of exact EFk allocations remained an intriguing open problem. We make notable progress towards this by proposing a simple, greedy, polynomial-time algorithm that computes EF2 allocations under budget constraints. Our algorithmic result implies the universal existence of EF2 allocations in this fair division context. The analysis of the algorithm exploits intricate structural properties of envy-freeness. Interestingly, the same algorithm also provides EF1 guarantees for important special cases. Specifically, we settle the existence of EF1 allocations for instances in which: (i) the value of each good is proportional to its size, (ii) all goods have the same size, or (iii) all the goods have the same value. Our EF2 result extends to the setting wherein the goods' sizes are agent specific.

翻译：我们研究在具有相同可加性估值但个体预算约束的智能体之间，对不可分割商品进行公平分配的问题。在此情境中，每个具有特定尺寸和价值的不可分割商品需要被分配，使得每个智能体获得的商品组合总尺寸不超过其预算。由于在不可分割商品情境中无嫉妒分配不一定存在，近年来诸如"至多k个商品的无嫉妒性"（EFk）等更具吸引力的松弛概念获得了广泛关注。在EFk分配中，每个智能体偏好自己的商品组合胜过其他任何智能体的组合（最多可去除k个商品），且智能体对未被分配的商品集（即慈善机构）同样具有有界嫉妒。吴等人（2021）近期证明，满足预算约束并最大化纳什社会福利的分配是1/4近似的EF1分配。然而，精确EFk分配的计算（甚至存在性）仍是一个引人入胜的开放问题。我们通过提出一种简单、贪心的多项式时间算法，在预算约束下计算EF2分配，为此取得了显著进展。该算法结果意味着在此公平分配场景中EF2分配普遍存在。算法的分析利用了无嫉妒性的精巧结构特性。有趣的是，该算法在重要特殊情况下同样提供EF1保证。具体而言，我们解决了以下实例中EF1分配的存在性：(i) 每个商品的价值与其尺寸成比例，(ii) 所有商品尺寸相同，或(iii) 所有商品价值相同。我们的EF2结果可扩展至商品尺寸因智能体而异的情境。