We study the problem of allocating indivisible goods among agents with additive valuation functions to achieve both fairness and efficiency under the constraint that each agent receives exactly the same number of goods (the \emph{balanced constraint}). While this constraint is common in real-world scenarios such as team drafts or asset division, it significantly complicates the search for allocations that are both fair and efficient. Envy-freeness up to one good (EF1) is a well-established fairness notion for indivisible goods. Pareto optimality (PO) and its stronger variant, fractional Pareto optimality (fPO), are widely accepted efficiency criteria. Our main contribution establishes both the existence and polynomial-time computability of allocations that are simultaneously EF1 and fPO under balanced constraints in two fundamental cases: (1) when each agent has a personalized bivalued valuation, and (2) when agents have at most two distinct valuation types,. Our algorithms leverage novel applications of maximum-weight matching in bipartite graphs and duality theory, providing the first polynomial-time solutions for these cases and offering new insights for constrained fair division problems.

翻译：我们研究了在可分割物品分配问题中，如何通过加性估值函数在满足每个代理获得相同数量物品（即平衡约束）的条件下，同时实现公平性与效率性。尽管该约束在团队选拔或资产分割等现实场景中普遍存在，但它显著增加了同时满足公平与效率的分配方案的求解难度。对于可分割物品，基于单物品嫉妒消除（EF1）的公平性概念已被广泛认可。帕累托最优（PO）及其强化版本——分数帕累托最优（fPO）则是普遍接受的效率性标准。我们的主要贡献在于证明了在以下两种基础情形中，同时满足EF1与fPO的平衡约束分配不仅存在，而且可在多项式时间内计算得到：（1）每个代理具有个性化二值估值函数；（2）代理至多拥有两种不同的估值类型。我们的算法创新性地运用了二分图最大权匹配与对偶理论，首次为这些情形提供了多项式时间解法，并为约束公平分割问题提供了新的理论视角。