Allocating $m$ indivisible goods among $n$ agents is a fundamental task in fair division. Recent work of Garg and Psomas [AAMAS 2025] initiated the study of parallel algorithms for envy-free up to one good (EF1) allocations, giving NC algorithms for $2$ and $3$ agents. They also showed CC-hardness results for simulating the classic Round Robin algorithm for EF1 allocations, even when each agent values at most $3$ goods and each good is valued by at most $3$ agents. We strengthen these results. For the case of $2$ agents, we quadratically improve the depth from $O(\log ^ 2 m) $ to $O(\log m)$ and the work from $O(m \log m)$ to $O(m)$. Furthermore, we significantly generalize beyond $3$ agents by giving NC algorithms for any constant number of agents. We also give randomized algorithms with depth $\tilde{O}(m/n)$ and polynomial work. As corollaries of these results, we obtain NC algorithms whenever each agent values at most $polylog(m)$ goods and each good is valued by at most $O(1)$ agents, and RNC algorithms when each agent values at most $polylog(m)$ goods. As such, our algorithms bypass the CC-hardness of Garg and Psomas by not simulating Round Robin. We also complement the aforementioned CC-hardness by showing the CC-completeness of simulating Round Robin. Lastly, beyond EF1 allocations, we show that computing envy-free up to $k$ goods allocations is possible for $k \approx \sqrt{m}$ in RNC, or $k = m^{\varepsilon}$ in sublinear depth for any constant $\varepsilon > 0$.

翻译：在公平分配领域，将$m$个不可分割物品分配给$n个$智能体是一项基础任务。Garg与Psomas的最新研究[AAMAS 2025]开创了无嫉妒至多一件物品(EF1)分配的并行算法研究，提出了面向$2$及$3$个智能体的NC算法。他们还证明，即使每个智能体最多评估$3$件物品且每件物品最多被$3$个智能体评估时，模拟经典Round Robin算法实现EF1分配仍存在CC困难性。我们强化了这些结论。针对两智能体情形，我们将深度从$O(\log ^ 2 m)$二次优化至$O(\log m)$，工作负载从$O(m \log m)$降至$O(m)$。此外，我们通过为任意常数个智能体提供NC算法，显著突破了三智能体限制。我们还提出了深度为$\tilde{O}(m/n)$且工作负载多项式级的有随机算法。作为这些结论的推论，我们在每个智能体最多评估$polylog(m)$件物品且每件物品最多被$O(1)$个智能体评估时获得NC算法，并在每个智能体最多评估$polylog(m)$件物品时获得RNC算法。因此，我们的算法通过不模拟Round Robin绕过了Garg与Psomas的CC困难性。我们还通过证明模拟Round Robin的CC完备性，补充了前述CC困难性结果。最后，超越EF1分配，我们证明在RNC中可实现$k \approx \sqrt{m}$时的无嫉妒至多$k$件物品分配，或在子线性深度内对任意常数$\varepsilon > 0$实现$k = m^{\varepsilon}$时的分配。