We study the complexity of a fundamental algorithm for fairly allocating indivisible items, the round-robin algorithm. For $n$ agents and $m$ items, we show that the algorithm can be implemented in time $O(nm\log(m/n))$ in the worst case. If the agents' preferences are uniformly random, we establish an improved (expected) running time of $O(nm + m\log m)$. On the other hand, assuming comparison queries between items, we prove that $\Omega(nm + m\log m)$ queries are necessary to implement the algorithm, even when randomization is allowed. We also derive bounds in noise models where the answers to queries are incorrect with some probability. Our proofs involve novel applications of tools from multi-armed bandit, information theory, as well as posets and linear extensions.

翻译：我们研究了公平分配不可分割物品的基本算法——轮询算法的复杂性。对于$n$个智能体和$m$个物品，我们证明该算法在最坏情况下可以在$O(nm\log(m/n))$时间内实现。如果智能体的偏好是均匀随机的，我们建立了改进的（期望）运行时间$O(nm + m\log m)$。另一方面，假设物品之间进行比较查询，我们证明即使允许随机化，实施该算法也需要$\Omega(nm + m\log m)$次查询。我们还在噪声模型中推导了界限，其中查询答案以一定概率不正确。我们的证明涉及多臂赌博机、信息论以及偏序集与线性扩展等工具的新应用。