Fair and Efficient Completion of Indivisible Goods

TL;DR

This work introduces the fair and efficient completion problem for indivisible goods with frozen resources, formalizing how to extend a partially preassigned allocation to a full one that satisfies fairness and efficiency. It analyzes the computational complexity of completing allocations under $EF1$, $Prop1$, and $MMS$, alone and in combination with $PO$, across additive valuations and its submodels binary additive and lexicographic; the authors deploy reductions from problems like Equitable Coloring and Rainbow Coloring, and, in binary additive cases, network-flow constructions to obtain positive results for threshold notions. The key findings include NP-hardness of $EF1$-Completion (even with two agents or with a Pareto-optimal frozen allocation in binary additive valuations) and NP-hardness of $EF1$-Completion and $Prop1$-Completion under general additive valuations, contrasted with polynomial-time solvability for $Prop1$ and $MMS$ completions in binary additive valuations and for $Prop1$+$PO$ and $MMS$ in lexicographic valuations; moreover, when the frozen allocation is $PO$, an $MMS$+$PO$ completion exists for binary additive valuations. The paper highlights a separation between envy-based fairness (hard) and threshold-based notions (tractable under restrictions), advances understanding of completion under fixed assignments, and lays groundwork for future work on MMS with $PO$, open cases, and broader resource models with frozen/forbidden constraints.

Abstract

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.