We formulate the problem of fair and efficient completion of indivisible goods, defined as follows: Given a partial allocation of indivisible goods among agents, does there exist an allocation of the remaining goods (i.e., a completion) that satisfies fairness and economic efficiency guarantees of interest? We study the computational complexity of the completion problem for prominent fairness and efficiency notions such as envy-freeness up one good (EF1), proportionality up to one good (Prop1), maximin share (MMS), and Pareto optimality (PO), and focus on the class of additive valuations as well as its subclasses such as binary additive and lexicographic valuations. We find that while the completion problem is significantly harder than the standard fair division problem (wherein the initial partial allocation is empty), the consideration of restricted preferences facilitates positive algorithmic results for threshold-based fairness notions (Prop1 and MMS). On the other hand, the completion problem remains computationally intractable for envy-based notions such as EF1 and EF1+PO even under restricted preferences.

翻译：我们提出了不可分割物品的公平高效补全问题，其定义如下：给定不可分割物品在智能体间的部分分配方案，是否存在对剩余物品的分配方案（即补全方案），能够满足特定的公平性与经济效率保证？我们针对若干重要的公平与效率概念——如单物品无嫉妒性（EF1）、单物品比例性（Prop1）、最大最小份额（MMS）和帕累托最优性（PO）——研究了补全问题的计算复杂性，并重点关注加性估值函数类及其子类（如二元加性估值和词典序估值）。研究发现，尽管补全问题的计算难度显著高于标准公平分配问题（即初始部分分配为空的情形），但在受限偏好假设下，基于阈值的公平性概念（Prop1和MMS）能够获得积极的算法结果。另一方面，对于基于嫉妒的公平性概念（如EF1及EF1+PO），即使考虑受限偏好，补全问题在计算上仍然难以处理。