We consider the classical fair division problem which studies how to allocate resources fairly and efficiently. We give a complete landscape on the computational complexity and approximability of maximizing the social welfare within (1) envy-free up to any item (EFX) and (2) envy-free up to one item (EF1) allocations of indivisible goods for both normalized and unnormalized valuations. We show that a partial EFX allocation may have a higher social welfare than a complete EFX allocation, while it is well-known that this is not true for EF1 allocations. Thus, our first group of results focuses on the problem of maximizing social welfare subject to (partial) EFX allocations. For $n=2$ agents, we provide a polynomial time approximation scheme (PTAS) and an NP-hardness result. For a general number of agents $n>2$, we present algorithms that achieve approximation ratios of $O(n)$ and $O(\sqrt{n})$ for unnormalized and normalized valuations, respectively. These results are complemented by the asymptotically tight inapproximability results. We also study the same constrained optimization problem for EF1. For $n=2$, we show a fully polynomial time approximation scheme (FPTAS) and complement this positive result with an NP-hardness result. For general $n$, we present polynomial inapproximability ratios for both normalized and unnormalized valuations. Our results also imply the price of EFX is $\Theta(\sqrt{n})$ for normalized valuations, which is unknown in the previous literature.

翻译：我们研究了经典的公平分配问题，即如何公平且高效地分配资源。我们完整刻画了在（1）任意物品无嫉妒（EFX）和（2）单物品无嫉妒（EF1）两种不可分割物品分配方案中，对于归一化与非归一化估值，最大化社会福利的计算复杂性与可近似性。研究表明，部分EFX分配可能比完全EFX分配具有更高的社会福利，而众所周知这一性质对EF1分配并不成立。因此，我们的第一组结果聚焦于在（部分）EFX分配约束下最大化社会福利的问题。对于$n=2$个智能体，我们给出了多项式时间近似方案（PTAS）和一个NP困难性结果。对于一般智能体数量$n>2$，我们提出了针对非归一化与归一化估值分别达到$O(n)$和$O(\sqrt{n})$近似比的算法。这些结果与渐近紧的不可近似性结果互为补充。我们还研究了EF1下的同一约束优化问题。对于$n=2$，我们展示了完全多项式时间近似方案（FPTAS），并用NP困难性结果补充了这一正面结论。对于一般$n$，我们给出了归一化与非归一化估值下的多项式不可近似比。我们的结果还揭示了归一化估值下EFX的价格为$\Theta(\sqrt{n})$，这一结论在先前文献中尚未被认知。