Improved Parallel Algorithms for EF1 Allocations
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914571910905856 |
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| author | Gowda, Kishen N Hershkowitz, D Ellis Huang, Richard Z Kehne, Gregory |
| author_facet | Gowda, Kishen N Hershkowitz, D Ellis Huang, Richard Z Kehne, Gregory |
| contents | 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 $\varepsilon > 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16791 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Improved Parallel Algorithms for EF1 Allocations Gowda, Kishen N Hershkowitz, D Ellis Huang, Richard Z Kehne, Gregory Data Structures and Algorithms Computer Science and Game Theory 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 $\varepsilon > 0$. |
| title | Improved Parallel Algorithms for EF1 Allocations |
| topic | Data Structures and Algorithms Computer Science and Game Theory |
| url | https://arxiv.org/abs/2605.16791 |