Improved Approximation of Weighted MMS Fairness for Indivisible Chores

We study how to fairly allocate a set of indivisible chores among n agents who may have different weights corresponding to their involvement in completing these chores. We found that some of the existing fairness notions may place agents with lower weights at a disadvantage, which motivates us to explore weighted maximin share fairness (WMMS). While it is known that a WMMS allocation may not exist, no non-trivial approximation has been discovered thus far. In this paper, we first design a simple sequential picking algorithm that solely relies on the agents’ ordinal rankings of the items, which achieves an approximation ratio of O(log n). Then, for the case involving two agents, we improve the approximation ratio to (√3+1)/2 ≈1.366, and prove that it is optimal. We also consider an online setting when the items arrive one after another and design an O(√n)-competitive online algorithm given the valuations are normalized