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：在公平分配中，将$m$个不可分割商品分配给$n$个智能体是一项基本任务。Garg与Psomas的最新研究[AAMAS 2025]开创了针对至多一个商品无嫉妒（EF1）分配的并行算法研究，给出了适用于$2$和$3$个智能体的NC算法。他们还证明了模拟经典轮询算法进行EF1分配是CC困难的，即使每个智能体至多看重$3个商品且每个商品至多被$3个智能体看重。我们强化了这些结果。针对$2个智能体情形，我们将深度从$O(\log ^ 2 m)$二次改进至$O(\log m)$，并将工作量从$O(m \log m)$优化至$O(m)$。此外，我们通过给出任意常数个智能体的NC算法，显著推广了$3个智能体以上的情形。我们还提出了深度为$\tilde{O}(m/n)$且工作量多项式复杂度的随机算法。作为这些结果的推论，我们得到了当每个智能体至多看重$polylog(m)$个商品且每个商品至多被$O(1)$个智能体看重时的NC算法，以及当每个智能体至多看重$polylog(m)$个商品时的RNC算法。因此，我们的算法通过避免模拟轮询机制绕过了Garg与Psomas的CC困难性。我们同时通过证明模拟轮询的CC完备性补充了前述CC困难结果。最后，超越EF1分配，我们证明了在RNC中可实现至多$k$个商品无嫉妒分配（$k \approx \sqrt{m}$），或在亚线性深度中实现$k = m^{\varepsilon}$（$\varepsilon > 0$为任意常数）。