We study the problem of fairly allocating indivisible goods (positively valued items) and chores (negatively valued items) among agents with decreasing marginal utilities over items. Our focus is on instances where all the agents have simple preferences; specifically, we assume the marginal value of an item can be either $-1$, $0$ or some positive integer $c$. Under this assumption, we present an efficient algorithm to compute leximin allocations for a broad class of valuation functions we call order-neutral submodular valuations. Order-neutral submodular valuations strictly contain the well-studied class of additive valuations but are a strict subset of the class of submodular valuations. We show that these leximin allocations are Lorenz dominating and approximately proportional. We also show that, under further restriction to additive valuations, these leximin allocations are approximately envy-free and guarantee each agent their maxmin share. We complement this algorithmic result with a lower bound showing that the problem of computing leximin allocations is NP-hard when $c$ is a rational number.

翻译：我们研究了在具有递减边际效用的主体间公平分配不可分割物品（正价值项）与家务（负价值项）的问题。我们的重点在于所有主体具有简单偏好的情形；具体而言，我们假设每个物品的边际价值可以是 $-1$、$0$ 或某个正整数 $c$。在此假设下，我们提出了一种高效算法，用于为称为序中性子模估值的一类广义估值函数计算勒克斯最小分配。序中性子模估值严格包含已被充分研究的可加估值类，但却是子模估值类的严格子集。我们证明这些勒克斯最小分配具有洛伦兹占优性和近似比例性。我们还证明，在进一步限制为可加估值时，这些勒克斯最小分配近似无嫉妒且能保证每个主体获得其最大最小份额。我们通过下界结果补充了该算法结论：当 $c$ 为有理数时，计算勒克斯最小分配问题是 NP 困难的。