This paper studies fair division of divisible and indivisible items among agents whose cardinal preferences are not necessarily monotone. We establish the existence of fair divisions and develop approximation algorithms to compute them. We address two complementary valuation classes, subadditive and nonnegative, which go beyond monotone functions. Considering both the division of cake (divisible resources) and allocation of indivisible items, we obtain fairness guarantees in terms of (approximate) envy-freeness (EF) and equability (EQ). In the context of envy-freeness, we prove that an EF division of a cake always exists under cake valuations that are subadditive and globally nonnegative. This result complements the nonexistence of EF allocations for burnt cakes known for more general valuations. In the indivisible-items setting, we establish the existence of EFE3 allocations for subadditive and globally nonnegative valuations. In addition, we obtain universal existence of EFE3 allocations under nonnegative valuations. We study equitability under nonnegative valuations. Here, we prove that EQE3 allocations always exist when the agents' valuations are nonnegative. Also, in the indivisible-items setting, we develop an approximation algorithm that, for given nonnegative valuations, finds allocations that are equitable within additive margins. Our results have combinatorial implications. For instance, the developed results imply the universal existence of proximately dense subgraphs: Given any graph $G=(V, E)$ and integer $k$ (at most $|V|$), there always exists a partition $V_1, V_2, \ldots, V_k$ of the vertex set such that the edge densities within the parts, $V_i$, are additively within four of each other. Further, such a partition can be computed efficiently.

翻译：本文研究在基数偏好不一定是单调的智能体之间对可分与不可分物品的公平分配问题。我们证明了公平分配的存在性，并开发了计算此类分配的近似算法。我们处理两个互补的估值类别：次可加和非负估值，这两类均超越了单调函数的范畴。针对蛋糕（可分资源）的划分和不可分物品的分配，我们获得了基于（近似）无嫉妒性（EF）和等值性（EQ）的公平性保证。在无嫉妒性框架下，我们证明对于满足次可加性和全局非负性的蛋糕估值，蛋糕的EF划分总是存在。这一结果补充了已知在更一般估值下"烧焦蛋糕"EF分配不存在的情形。在不可分物品设定中，我们证明了对于次可加且全局非负的估值，EFE3分配的存在性。此外，我们获得了在非负估值下EFE3分配的普遍存在性。我们研究了非负估值下的等值性。在此，我们证明了当智能体估值非负时，EQE3分配总是存在。同时，在不可分物品设定中，我们开发了一种近似算法，对于给定的非负估值，该算法能找到在加性误差范围内满足等值性的分配。我们的结果具有组合学意义。例如，所发展的结果意味着近似稠密子图的普遍存在性：给定任意图$G=(V, E)$和整数$k$（至多$|V|$），总存在顶点集的一个划分$V_1, V_2, \ldots, V_k$，使得各部分$V_i$内部的边密度在加性误差四以内相互接近。此外，这种划分可以被高效计算。