Online Fair Division Meets Reordering Buffers

We study the _online fair division_ of indivisible _mixed manna_ among agents with additive valuation functions. Under the standard online model, at each time step an indivisible item arrives; each agent may assign it a positive, negative, or zero value, and it must be irrevocably allocated, before the arrival of the next item. At the same time, we also wish to maintain some fairness guarantee, and in this work we focus on _envy-freeness_ (EF) and one of its most prominent relaxations, _envy-freeness up to one item_ (EF1). Given the strong negative and the scarce positive results for this problem without additional assumptions, we augment our algorithms with _buffers_ that can store and rearrange a limited number of items. This setting interpolates naturally between the fully online case (no buffer) and the fully offline case (a buffer large enough to hold all items). We show that algorithms equipped with reasonably sized buffers can achieve strong guarantees for personalized $k$-value instances, i.e., instances in which each agent assigns at most $k$ distinct values to items. In particular, we construct allocations that are EF1 at every time step and EF at most time steps, using a buffer of size linear in $k$ and in the number of agents. Our approach relies on novel combinatorial arguments and on constructing a sequence of envy-free matchings that allocates most items. Finally, we extend our results to general additive valuation functions, with a dependence on the largest per-agent ratio between two values of the same sign, and we also identify limitations of our approach via impossibility results on the use of buffers with smaller size.