A Fair Allocation is Approximately Optimal for Indivisible Chores, or Is It?

In this paper, we study the allocation of indivisible chores and consider the problem of finding a fair allocation that is approximately efficient. We shift our attention from the multiplicative approximation to the additive one. Our results are twofold, with (1) bounding how the optimal social cost escalates resulting from fairness requirements and (2) presenting the hardness of approximation for the problems of finding fair allocations with the minimum social cost. To quantify the escalation, we introduce cost of fairness (CoF) $\unicode{x2014}$ an alternative to the price of fairness (PoF) $\unicode{x2014}$ to bound the difference (v.s. ratio for PoF) between the optimal social cost with and without fairness constraints in the worst-case instance. We find that CoF is more informative than PoF for chores in the sense that the PoF is infinity regarding all EQX (equitable up to any item), EQ1 (equitable up to one item) and EF1 (envy-free up to one item), while the CoF is $n$ regarding EQX and 1 regarding EQ1 and EF1, where $n$ is the number of agents. For inapproximability, we present a detailed picture of hardness of approximation. We prove that finding the optimal EQX allocation within an additive approximation factor of $n$ is NP-hard for any $n \geq 2$ where $n$ is the number of agents and the cost functions are normalized to 1. For EQ1 and EF1, the problem is NP-hard when the additive factor is a constant and $n \geq 3$. When $n = 2$, we design additive approximation schemes for EQ1 and EF1.