We consider the computation for allocations of indivisible chores that are approximately EFX and Pareto optimal (PO). Recently, Garg et al. (2024) show the existence of $3$-EFX and PO allocations for bi-valued instances, where the cost of an item to an agent is either $1$ or $k$ (where $k > 1$) by rounding the (fractional) earning restricted equilibrium. In this work, we improve the approximation ratio to $(2-1/k)$, while preserving the Pareto optimality. Instead of rounding fractional equilibrium, our algorithm starts with the integral EF1 equilibrium for bi-valued chores, introduced by Garg et al. (AAAI 2022) and Wu et al. (EC 2023), and reallocates items until approximate EFX is achieved. We further improve our result for the case when $k=2$ and devise an algorithm that computes EFX and PO allocations.

翻译：我们研究不可分割家务分配的近似无嫉妒性（EFX）与帕累托最优（PO）计算问题。近期，Garg等人（2024）通过舍入（分数型）收益限制均衡，证明了在双值实例（物品对智能体的成本仅为1或k，其中k>1）中存在3-EFX且PO的分配。本工作中，我们在保持帕累托最优性的同时，将近似比提升至(2-1/k)。与舍入分数均衡不同，我们的算法从Garg等人（AAAI 2022）与Wu等人（EC 2023）提出的双值家务积分EF1均衡出发，通过物品重分配实现近似EFX。针对k=2的特殊情形，我们进一步提出可计算EFX且PO分配的改进算法。