The Task Completion Problem and its Application to Crash-Resilient Computation

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 $\alpha n$ nodes may crash for some constant $\alpha<1$. 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 $\alpha n$ 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.