Almost (Weighted) Proportional Allocations for Indivisible Chores

In this paper, we study how to fairly allocate a set of indivisible chores to a number of (asymmetric) agents with additive cost functions. We consider the fairness notion of (weighted) proportionality up to any item (PROPX), and show that a (weighted) PROPX allocation always exists and can be computed efficiently. We also consider the partial information setting, where the algorithms can only use agents' ordinal preferences. We design algorithms that achieve 2-approximate (weighted) PROPX, and the approximation ratio is optimal. We complement the algorithmic results by investigating the relationship between (weighted) PROPX and other fairness notions such as maximin share and AnyPrice share, and bounding the social welfare loss by enforcing the allocations to be (weighted) PROPX.