Complexity of Round-Robin Allocation with Potentially Noisy Queries

TL;DR

This paper analyzes the computational complexity and query requirements of the round-robin algorithm for fair allocation, providing worst-case, average-case, and noisy preference models with novel theoretical bounds and tools.

Contribution

It offers new complexity bounds for implementing and querying the round-robin algorithm under various models, including noisy preferences and random agent preferences.

Findings

Worst-case implementation time is O(nm log(m/n)).

Expected running time improves to O(nm + m log m) under random preferences.

Query complexity lower bound is Ω(nm + m log m) even with randomization.

Abstract

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.