We study two-sided matching markets under hereditary constraints, which extend beyond simple capacity limits and arise in applications such as diversity requirements and refugee resettlement. In these settings, fairness and non-wastefulness are often incompatible, and existing approaches typically address this tension by prioritizing one property at the expense of the other. We take a different approach by relaxing both properties simultaneously in a controlled and symmetric manner. We introduce two notions indexed by an integer $k$: envy-received up to $k$ peers (ER-$k$) and non-wastefulness up to $k$ objections (NW-$k$). Our main theoretical result shows that ER-$k$ and NW-$k$ are always compatible under hereditary constraints for any fixed $k$. We provide two equivalent polynomial-time algorithms to compute such matchings: a $k$-admissible cutoff algorithm and a $k$-admissible college-proposing deferred acceptance mechanism. Finally, experimental results demonstrate that even small relaxations achieve a favorable balance between fairness and non-wastefulness.

翻译：我们研究了遗传约束下的双边匹配市场，这类约束超越了简单的容量限制，并出现在多样性要求和难民重新安置等应用中。在这些设定下，公平性与非浪费性往往不可兼得，现有方法通常通过优先满足某一性质而牺牲另一性质来处理这一矛盾。我们采取一种不同的方法，以受控且对称的方式同时松弛这两个性质。我们引入了由整数$k$索引的两个概念：至多$k$个同伴的接收嫉妒（ER-$k$）和至多$k$个反对的非浪费性（NW-$k$）。我们的主要理论结果表明，对于任意固定的$k$，ER-$k$和NW-$k$在遗传约束下总是兼容的。我们提供了两种等价的多项式时间算法来计算此类匹配：一种$k$可接受截止算法和一种$k$可接受学院提议延迟接受机制。最后，实验结果表明，即使是较小的松弛也能在公平性与非浪费性之间实现良好的平衡。