We revisit the setting of fair allocation of indivisible items among agents with heterogeneous, non-monotone valuations. We explore the existence and efficient computation of allocations that approximately satisfy either envy-freeness or equity constraints. Approximate envy-freeness ensures that each agent values her bundle at least as much as those given to the others, after some (or any) item removal, while approximate equity guarantees roughly equal valuations among agents, under similar adjustments. As a key technical contribution of this work, by leveraging fixed-point theorems (such as Sperner's Lemma and its variants), we establish the existence of {\em envy-free-up-to-one-good-and-one-chore} ($\text{EF1}^c_g$) and {\em equitable-up-to-one-good-and-one-chore} ($\text{EQ1}^c_g$) allocations, for non-monotone valuations that are always either non-negative or non-positive. These notions represent slight relaxations of the well-studied {\em envy-free-up-to-one-item} (EF1) and {\em equitable-up-to-one-item} (EQ1) guarantees, respectively. Our existential results hold even when items are arranged in a path and bundles must form connected sub-paths. The case of non-positive valuations, in particular, has been solved by proving a novel multi-coloring variant of Sperner's Lemma that constitutes a combinatorial result of independent interest. In addition, we also design a polynomial-time dynamic programming algorithm that computes an $\text{EQ1}^c_g$ allocation. For monotone non-increasing valuations and path-connected bundles, all the above results can be extended to EF1 and EQ1 guarantees as well. Finally, we provide existential and computational results for certain stronger {\em up-to-any-item} equity notions under objective valuations, where items are partitioned into goods and chores.

翻译：我们重新审视具有异质非单调估值的智能体之间不可分物品的公平分配问题。我们探讨近似满足无嫉妒性或公平性约束的分配方案的存在性与高效计算。近似无嫉妒性确保每个智能体在移除某些（或任意）物品后，对自己所获束的估值不低于对其他智能体所获束的估值；而近似公平性则保证在类似调整下智能体间的估值大致相等。作为本工作的核心技术贡献，通过运用不动点定理（如Sperner引理及其变体），我们证明了对于始终非负或非正的非单调估值，存在{\em 至多移除一件物品与一项杂务的无嫉妒分配}（$\text{EF1}^c_g$）和{\em 至多移除一件物品与一项杂务的公平分配}（$\text{EQ1}^c_g$）。这些概念分别是对已深入研究的{\em 至多移除一件物品的无嫉妒性}（EF1）和{\em 至多移除一件物品的公平性}（EQ1）保证的轻微松弛。即使物品按路径排列且分配束必须形成连通子路径，我们的存在性结果仍然成立。特别地，针对非正估值的情况，我们通过证明Sperner引理的一个新颖多着色变体得以解决，该变体本身构成一个具有独立意义的组合学结果。此外，我们还设计了一个多项式时间动态规划算法来计算$\text{EQ1}^c_g$分配。对于单调非增估值及路径连通分配束，上述所有结果均可扩展至EF1和EQ1保证。最后，我们针对客观估值下某些更强的{\em 至多任意物品}公平性概念提供了存在性与计算性结果，其中物品被划分为正品与杂务。