We study the problem of fairly allocating indivisible goods and chores under category constraints. Specifically, there are $n$ agents and $m$ indivisible items which are partitioned into categories with associated capacities. An allocation is considered feasible if each bundle satisfies the capacity constraints of its respective categories. For the case of two agents, Shoshan et al. (2023) recently developed a polynomial-time algorithm to find a Pareto-optimal allocation satisfying a relaxed version of envy-freeness, called EF$[1,1]$. Extending such guarantees beyond two agents has remained open. We make progress toward this question by proving that for any number $n$ of agents, there always exists a Pareto-optimal allocation in which each agent can be made envy-free by reallocating at most $\min \{k+1,n\}(n-1)$ items. Moreover, when the number of agents is constant, we give a polynomial-time algorithm to compute such an allocation. Our results rely on a new application of the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma to a simplex of agent weights, which may be of independent interest for fair division problems involving indivisible items.

翻译：我们研究了在类别约束下公平分配不可分割物品与杂务的问题。具体而言，存在$n$个智能体和$m$个不可分割物品，这些物品被划分为若干类别，并附带相应容量限制。若每个束满足其所属类别的容量约束，则该分配被视为可行。针对两个智能体的情形，Shoshan等人（2023）近期提出了一种多项式时间算法，可找到满足松弛版本无嫉妒性（称为EF$[1,1]$）的帕累托最优分配。将该保障扩展至超过两个智能体的问题仍未解决。我们通过证明以下结论推动该问题的进展：对任意$n$个智能体，总存在一个帕累托最优分配，使得每个智能体可通过重新分配至多$\min \{k+1,n\}(n-1)$个物品实现无嫉妒性。此外，当智能体数量为常数时，我们给出了计算此类分配的多项式时间算法。我们的结果依赖于Knaster-Kuratowski-Mazurkiewicz（KKM）引理在智能体权重单纯形上的新应用，这一方法可能对涉及不可分割物品的公平分配问题具有独立参考价值。