We study the problem of fairly allocating a set of indivisible goods to a set of $n$ agents with additive valuation functions. We focus on the very demanding notion of \textit{groupwise maximin share fairness} (GMMS), which requires that each agent $i$ receives value comparable to their maximin share, where the latter is computed \textit{with respect to any subset of agents that contains $i$}. We show that it is possible to compute $(φ-1)$-approximate GMMS allocations in polynomial time, where $φ\approx 1.618$ is the golden ratio). This improves on the previously known guarantee of $4/7$ of Chaudhury et al. [SICOMP; 2021] and Amanatidis et al. [TCS; 2020]. We propose a simple algorithm that maintains the same main properties as the Draft-and-Eliminate algorithm of Amanatidis et al. [TCS, 2020] and we improve on the approximation guarantee analysis by carefully bounding the relevant value within any subinstance induced by the restriction of our allocation to a subset of agents. Our analysis is asymptotically tight for algorithms that share these properties and has the additional benefit of giving improved guarantees for restricted settings; in particular, when the agents agree on the top $n$ goods or when the number of agents is small. To illustrate the challenges of going beyond the guarantees of our algorithm, we also present a variant with an improved approximation of $(\sqrt{10}-1)/3 \approx 0.72$ for the case of three agents. To achieve this improvement we partially characterize the maximin share guarantees of short picking sequences for a small number of goods.

翻译：我们研究将一组不可分割物品公平分配给一组具有可加估值函数的$n$个智能体的问题。我们聚焦于非常严格的概念——**群体最大最小份额公平性**（GMMS），该要求每个智能体$i$获得的价值须与其最大最小份额相当，而后者是**针对任何包含$i$的智能体子集**计算得出的。我们证明可在多项式时间内计算$(φ-1)$-近似GMMS分配，其中$φ\approx 1.618$为黄金比例。这改进了Chaudhury等人[SICOMP; 2021]和Amanatidis等人[TCS; 2020]此前已知的$4/7$保证。我们提出一种简单算法，该算法保留了Amanatidis等人[TCS, 2020]的“草拟与消除”算法的主要性质，并通过谨慎界定由分配限制至某个智能体子集所诱导的任何子实例中的相关值，改进了近似保证的分析。我们的分析对于具有这些性质的算法是渐近紧的，且额外优势是为受限场景（特别是当智能体对前$n$个物品达成一致或智能体数量较少时）提供了改进保证。为说明超越我们算法保证的挑战，我们还针对三个智能体的情况提出一种变体，其近似比提升至$(\sqrt{10}-1)/3 \approx 0.72$。为实现这一改进，我们部分刻画了小数量物品短选取序列的最大最小份额保证。