Online MMS Allocation for Chores
Jiaxin Song, Biaoshuai Tao, Wenqian Wang, Yuhao Zhang

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 agents in an online setting, where items arrive sequentially and must be allocated irrevocably upon arrival. The goal is to produce an -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 -MMS guarantee remains the best known, while the strongest lower bound is still only . We close this gap on the negative side by proving that for any fixed and , no algorithm can guarantee an -MMS allocation. Notably, this lower bound holds precisely for every , without hiding…
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