We study fair division of $m$ indivisible chores among $n$ agents with additive preferences. We consider the desirable fairness notions of envy-freeness up to any chore (EFX) and envy-freeness up to $k$ chores (EF$k$), alongside the efficiency notion of Pareto optimality (PO). We present the first constant approximations of these notions, showing the existence of: - 5-EFX allocations, which improves the best-known factor of $O(n^2)$-EFX. - 3-EFX and PO allocations for the special case of bivalued instances, which improves the best-known factor of $O(n)$-EFX without any efficiency guarantees. - 2-EF2 + PO allocations, which improves the best-known factor of EF$m$ and PO. A notable contribution of our work is the introduction of the novel concept of earning-restricted (ER) competitive equilibrium for fractional allocations, which limits agents' earnings from each chore. Technically, our work addresses two main challenges: proving the existence of an ER equilibrium and designing algorithms that leverage ER equilibria to achieve the above results. To tackle the first challenge, we formulate a linear complementarity problem (LCP) formulation that captures all ER equilibria and show that the classic complementary pivot algorithm on the LCP must terminate at an ER equilibrium. For the second challenge, we carefully set the earning limits and use properties of ER equilibria to design sophisticated procedures that involve swapping and merging bundles to meet the desired fairness and efficiency criteria. We expect that the concept of ER equilibrium will be instrumental in deriving further results on related problems.

翻译：我们研究了在具有加性偏好的$n$个智能体之间公平分配$m$项不可分割杂务的问题。我们考虑了理想的公平性概念：对任意杂务无嫉妒（EFX）和对$k$项杂务无嫉妒（EF$k$），以及帕累托最优（PO）的效率概念。我们首次提出了这些概念的常数近似结果，证明了以下分配的存在性：- 5-EFX分配，这改进了先前已知的$O(n^2)$-EFX最佳因子。- 针对二值化实例的特殊情况，存在3-EFX且PO的分配，这改进了先前已知的$O(n)$-EFX最佳因子（且无任何效率保证）。- 2-EF2 + PO分配，这改进了先前已知的EF$m$与PO的最佳因子。我们工作的一个显著贡献是引入了收益受限（ER）竞争均衡这一新颖概念，用于分数分配，它限制了智能体从每项杂务中获得的收益。在技术上，我们的工作解决了两个主要挑战：证明ER均衡的存在性，以及设计利用ER均衡来实现上述结果的算法。为应对第一个挑战，我们构建了一个线性互补问题（LCP）公式来刻画所有ER均衡，并证明了在该LCP上经典的互补转轴算法必定终止于一个ER均衡。对于第二个挑战，我们精心设置收益限制，并利用ER均衡的性质设计了涉及交换与合并捆绑包的复杂流程，以满足所需的公平性和效率标准。我们预期ER均衡的概念将在相关问题的进一步研究中发挥重要作用。