Reforming an Unfair Allocation by Exchanging Goods
Sheung Man Yuen, Ayumi Igarashi, Naoyuki Kamiyama, Warut Suksompong
TL;DR
This work studies reforming an existing allocation of indivisible goods to EF1 via pairwise exchanges that preserve bundle sizes. It maps out a detailed complexity landscape across the number of agents and utility structures, showing polynomial-time solvability in several key cases (e.g., two agents with identical utilities, constant-number binary utilities) and proving NP-hardness in most other scenarios, including strong NP-hardness for identical utilities with many agents. It also establishes that a balanced size vector guarantees EF1 existence and derives essentially tight worst-case bounds on the required number of exchanges, with precise constants for several utility families. The results offer both algorithmic procedures (e.g., dynamic programming, greedy exchange strategies) and hardness proofs that illuminate the computational limits of reformatting allocations toward fairness in practice. Overall, the paper advances the understanding of how initial unfair allocations can be reformed efficiently or where inherent complexity blocks such reform, with implications for personnel, inventory, and exhibit allocations under fairness constraints.
Abstract
Fairly allocating indivisible goods is a frequently occurring task in everyday life. Given an initial allocation of the goods, we consider the problem of reforming it via a sequence of exchanges to attain fairness in the form of envy-freeness up to one good (EF1). We present a vast array of results on the complexity of determining whether it is possible to reach an EF1 allocation from the initial allocation and, if so, the minimum number of exchanges required. In particular, we uncover several distinctions based on the number of agents involved and their utility functions. Furthermore, we derive essentially tight bounds on the worst-case number of exchanges needed to achieve EF1.