We study the problem of fair allocation of chores to agents with additive preferences. In the discrete setting, envy-freeness up to any chore (EFX) has emerged as a compelling fairness criterion. However, establishing its (non-)existence or achieving a meaningful approximation remains a major open question. The current best guarantee is the existence of $O(n^2)$-EFX allocations for $n$ agents, obtained through a sophisticated algorithm (Zhou and Wu, 2022). In this paper, we show the existence of $4$-EFX allocations, providing the first constant-factor approximation of EFX. We also investigate the existence of allocations that are both fair and efficient, using Pareto optimality (PO) as our efficiency criterion. For the special case of bivalued instances, we establish the existence of allocations that are both $3$-EFX and PO, thus improving the current best factor of $O(n)$-EFX without any efficiency guarantees. For general additive instances, the existence of allocations that are $\alpha$-EF$k$ and PO has remained open for any constant values of $\alpha$ and $k$, where EF$k$ denotes envy-freeness up to $k$ chores. We provide the first positive result in this direction by showing the existence of allocations that are $2$-EF$2$ and PO. Our results are obtained via a novel economic framework called earning restricted (ER) competitive equilibrium for fractional allocations, which limits agents' earnings from each chore. We show the existence of ER equilibria by formulating it as an linear complementarity problem (LCP) and proving that the classic complementary pivot algorithm on the LCP terminates at an ER equilibrium. We design algorithms that carefully round fractional ER equilibria, and perform bundle swaps and merges to meet the desired fairness and efficiency criteria. We expect that the concept of ER equilibrium will be useful in deriving further results on related problems.

翻译：我们研究了具有加性偏好的智能体之间杂务的公平分配问题。在离散设定下，任意杂务无嫉妒性（EFX）已成为一个引人注目的公平性标准。然而，确定其（不）存在性或实现有意义的近似仍是一个主要的开放性问题。目前的最佳保证是通过复杂算法（Zhou and Wu, 2022）获得的$n$个智能体存在$O(n^2)$-EFX分配。在本文中，我们证明了$4$-EFX分配的存在性，首次提供了EFX的恒定因子近似。我们还研究了同时满足公平性和效率性的分配的存在性，以帕累托最优性（PO）作为效率标准。对于双值实例的特殊情况，我们证明了同时满足$3$-EFX和PO的分配的存在性，从而改进了当前无任何效率保证的$O(n)$-EFX最佳因子。对于一般加性实例，对于任何常数$\alpha$和$k$，同时满足$\alpha$-EF$k$和PO的分配的存在性一直是一个开放问题，其中EF$k$表示最多$k$个杂务的无嫉妒性。我们通过证明$2$-EF$2$和PO分配的存在性，首次在这一方向上提供了积极结果。我们的结果是通过一种称为收益受限（ER）竞争均衡的新颖经济框架获得的，该框架用于分数分配，限制了智能体从每个杂务中获得的收益。我们通过将其表述为一个线性互补问题（LCP），并证明LCP上的经典互补转轴算法终止于一个ER均衡，从而证明了ER均衡的存在性。我们设计了算法，仔细地对分数ER均衡进行舍入，并通过捆绑交换与合并来满足所需的公平性和效率性标准。我们预期ER均衡的概念将在相关问题的进一步结果推导中发挥重要作用。