Improved Parallel Algorithms for EF1 Allocations

2026-05-15

AI Summary

This work addresses the parallel complexity of achieving EF1 fairness in indivisible goods allocation. Bypassing the CC-completeness barrier inherent in simulating Round Robin, it introduces novel NC and RNC parallel algorithms. Under the assumptions of a constant number of agents and sparse valuations, the algorithm for two agents improves depth from O(log²m) to O(logm) with O(m) total work. The study further presents the first deterministic NC algorithm for any constant number of agents and extends the approach to EFk fairness, yielding an RNC solution with sublinear depth when k ≈ √m. By integrating combinatorial optimization with complexity theory, this work substantially enhances the parallel efficiency of computing EF1 and EFk allocations.

Abstract

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 $\varepsilon > 0$.

Problem

• fair division
• EF1 allocations
• parallel algorithms
• indivisible goods
• computational complexity

Innovation

• parallel algorithms
• EF1 allocations
• NC complexity
• fair division
• randomized algorithms