Online MMS Allocation for Chores

TL;DR

This paper investigates the online fair division of indivisible chores among agents, establishing fundamental impossibility results and proposing algorithms that guarantee approximate MMS allocations under various conditions.

Contribution

It proves tight lower bounds for online MMS allocation and introduces algorithms achieving bounded approximation ratios based on problem parameters.

Findings

No algorithm can guarantee an $(n - ext{small constant})$-MMS allocation.

An online algorithm guarantees an $O( ext{max of } n, k, ext{log } D)$-MMS allocation.

For bi-valued cases with at most two disutilities per agent, the algorithm guarantees a 3.7-MMS allocation.

Abstract

We study the problem of fair division of indivisible chores among $n$ agents in an online setting, where items arrive sequentially and must be allocated irrevocably upon arrival. The goal is to produce an $\alpha$-MMS allocation at the end. Several recent works have investigated this model, but have only succeeded in obtaining non-trivial algorithms under restrictive assumptions, such as the two-agent bi-valued special case (Wang and Wei, 2025), or by assuming knowledge of the total disutility of each agent (Zhou, Bai, and Wu, 2023). For the general case, the trivial $n$-MMS guarantee remains the best known, while the strongest lower bound is still only $2$. We close this gap on the negative side by proving that for any fixed $n$ and $\varepsilon$, no algorithm can guarantee an $(n - \varepsilon)$-MMS allocation. Notably, this lower bound holds precisely for every $n$, without hiding constants in big-$O$ notation, thereby exactly matching the trivial upper bound. Despite this strong impossibility result, we also present positive results. We provide an online algorithm that applies in the general case, guaranteeing a $\min\{n, O(k), O(\log D)\}$-MMS allocation, where $k$ is the maximum number of distinct disutilities across all agents and $D$ is the maximum ratio between the largest and smallest disutilities for any agent. This bound is reasonable across a broad range of scenarios and, for example, implies that we can achieve an $O(1)$-MMS allocation whenever $k$ is constant. Moreover, to optimize the constant in the important personalized bi-valued case, we show that if each agent has at most two distinct disutilities, our algorithm guarantees a $(2 + \sqrt{3}) \approx 3.7$-MMS allocation.