We study the Task Completion problem, in which $M$ abstract tasks must be completed by a network of $n$ crash-prone nodes, where up to $αn$ nodes may crash for some constant $α<1$. Our main result is a deterministic congested-clique algorithm that completes all $M$ tasks in $O(\lceil M/n\rceil \log n)$ rounds. This round complexity is optimal up to $\log\log n$ terms. The key technical ingredient underlying our algorithm is a novel combinatorial structure, which we call a \emph{load balancing covering family}. In essence, this covering family induces, for each task, a subset of nodes responsible for attempting to complete it. The properties of the load balancing covering family guarantee that, regardless of which tasks remain incomplete and which nodes crash, (i) no node is overloaded with incomplete tasks, and (ii) no task is left with too few potential assigned nodes. This yields a balanced per-node workload and prevents non-crashed nodes from being concentrated on a small subset of tasks, thereby ensuring sufficient progress in completing the remaining tasks. As an application of our task completion method, we give a deterministic algorithm for simulating any $T$-round congested-clique algorithm in the presence of up to $αn$ crash faults in $O(T^2 \log n + T \log^2 n)$ rounds. This improves upon a recent result by Censor-Hillel et al. (DISC~2025), which requires $T^2\cdot 2^{O(\sqrt{\log n}\log\log n)}$ rounds.

翻译：我们研究任务完成问题，其中 $M$ 个抽象任务需由 $n$ 个易崩溃节点组成的网络完成，对于某个常数 $\alpha<1$，至多 $\alpha n$ 个节点可能崩溃。我们的主要结果是一种确定性的拥塞-团算法，能在 $O(\lceil M/n\rceil \log n)$ 轮内完成所有 $M$ 个任务。该轮复杂度在 $\log\log n$ 项内达到最优。算法背后的关键技术成分是一种新颖的组合结构，我们称之为\emph{负载均衡覆盖族}。本质上，该覆盖族为每个任务诱导一个负责尝试完成该任务的节点子集。负载均衡覆盖族的性质保证，无论哪些任务未完成以及哪些节点崩溃，(i) 没有节点会因未完成任务而过载，且 (ii) 每个任务不会剩下过少的潜在分配节点。这产生了均衡的每节点工作负载，并防止未崩溃节点集中于少量任务上，从而确保在完成剩余任务时取得足够进展。作为任务完成方法的一个应用，我们给出一种确定性算法，用于在至多 $\alpha n$ 个崩溃故障存在时，模拟任意 $T$ 轮的拥塞-团算法，其轮复杂度为 $O(T^2 \log n + T \log^2 n)$。这改进了 Censor-Hillel 等人 (DISC~2025) 的近期结果，该结果需要 $T^2\cdot 2^{O(\sqrt{\log n}\log\log n)}$ 轮。