In this work, we revisit well-studied problems of fair allocation of indivisible items among agents with general, non-monotone valuations. We explore the existence and efficient computation of allocations that satisfy either fairness or equity constraints. The fairness notions we consider ensure that each agent values her bundle at least as much as others', allowing for (any or some) item removal, while the equity guarantees roughly equal valuations among agents, with similar adjustments. For objective valuations where items are classified as either goods or chores, we present a pseudo-polynomial local-search algorithm computing an ``equitable-up-to-any-good-or-any-chore'' (EQX*) allocation, a weaker version of an ``equitable-up-to-any-item" (EQX) allocation. Additionally, we provide a polynomial-time greedy algorithm that computes an ``equitable-up-to-one-item" (EQ1) allocation, and a similar algorithm returning an EQX* allocation when the valuations are also additive. 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 ``equitable-up-to-one-good-and-one-chore'' (EQ1*) and ``envy-free-up-to-one-good-and-one-chore'' (EF1*) allocations for non-negative (and possibly non-objective and non-monotone) valuations. This holds even when items are arranged in a path and bundles must form connected sub-paths. Additionally, we present a polynomial-time dynamic-programming algorithm that computes an EQ1* allocation. Finally, we extend the EF1* and EQ1* results to non-positive valuations using a novel multi-coloring variant of Sperner's lemma, a combinatorial result of independent interest. For monotone non-increasing valuations and path-connected bundles, this implies the existence of EF1 and EQ1 allocations, with EQ1 allocations being efficiently computable.

翻译：本研究重新审视了具有一般非单调估值的主体间不可分物品公平分配的经典问题。我们探讨了满足公平性或均等性约束的分配方案的存在性与高效计算。所考虑的公平性概念确保每个主体对其所获束的估值不低于其他主体（允许移除任意或某些物品），而均等性则保证主体间估值大致相等，并允许类似调整。对于物品被分类为商品或杂务的客观估值情形，我们提出了一种伪多项式局部搜索算法，用于计算"除任意商品或任意杂务外均等"（EQX*）分配——这是"除任意物品外均等"（EQX）分配的弱化版本。此外，我们提供了一种多项式时间贪心算法来计算"除一件物品外均等"（EQ1）分配，以及当估值具有可加性时能返回EQX*分配的类似算法。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），我们证明了对于非负（可能非客观且非单调）估值，存在"除一件商品和一件杂务外均等"（EQ1*）与"除一件商品和一件杂务外无嫉妒"（EF1*）分配。这一结论甚至在物品按路径排列且分配束必须形成连通子路径时依然成立。此外，我们提出了一种计算EQ1*分配的多项式时间动态规划算法。最后，通过Sperner引理的新型多着色变体（这一组合结果具有独立理论价值），我们将EF1*与EQ1*的结果推广至非正估值情形。对于单调非递增估值及路径连通束，这意味着EF1与EQ1分配的存在性，且EQ1分配可被高效计算。