Forward-backward Contention Resolution Schemes for Fair Rationing
Will Ma, Calum MacRury, Cliff Stein
TL;DR
The paper develops forward-backward contention resolution schemes (FB-CRS) for online rationing under uncertain demands, bridging CRS with ex-ante fairness guarantees for two primary feasibility constraints: rank-1 matroids and knapsack. It introduces a reduction framework from rationing targets to an ex-ante feasible region via inverse CDF quantiles, enabling CRS-based online policies that preserve per-agent service guarantees. The authors prove a tight two-order FB-CRS bound for rank-1 matroids of $1/(1+e^{-1/2}) 2 ext{ approx }0.622$, and a two-order FB-CRS bound for knapsack of $1/3$, with an upper bound of $1/(2+e^{-1}) 2 ext{ approx }0.422$; these results improve prior single-order and random-order prophet inequalities and extend the applicability of CRS to online rationing with forward-backward arrivals. The technical core hinges on single-unit CRS analyses, phi-based constructions, and knapsack-invariant arguments to ensure feasibility and near-optimal guarantees, contributing to both theory and practical online resource allocation contexts such as food-bank logistics.
Abstract
We use contention resolution schemes (CRS) to derive algorithms for the fair rationing of a single resource when agents have stochastic demands. We aim to provide ex-ante guarantees on the level of service provided to each agent, who may measure service in different ways (Type-I, II, or III), calling for CRS under different feasibility constraints (rank-1 matroid or knapsack). We are particularly interested in two-order CRS where the agents are equally likely to arrive in a known forward order or its reverse, which is motivated by online rationing at food banks. In particular, we derive a two-order CRS for rank-1 matroids with guarantee $1/(1+e^{-1/2})\approx 0.622$, which we prove is tight. This improves upon the $1/2$ guarantee that is best-possible under a single order (Alaei, SIAM J. Comput. 2014), while achieving separation with the $1-1/e\approx 0.632$ guarantee that is possible for random-order CRS (Lee and Singla, ESA 2018). Because CRS guarantees imply prophet inequalities, this also beats the two-order prophet inequality with ratio $(\sqrt{5}-1)/2\approx 0.618$ from (Arsenis, SODA 2021), which was tight for single-threshold policies. Rank-1 matroids suffice to provide guarantees under Type-II or III service, but Type-I service requires knapsack. Accordingly, we derive a two-order CRS for knapsack with guarantee $1/3$, improving upon the $1/(3+e^{-2})\approx 0.319$ guarantee that is best-possible under a single order (Jiang et al., SODA 2022). To our knowledge, $1/3$ provides the best-known guarantee for knapsack CRS even in the offline setting. Finally, we provide an upper bound of $1/(2+e^{-1})\approx 0.422$ for two-order knapsack CRS, strictly smaller than the upper bound of $(1-e^{-2})/2\approx0.432$ for random-order knapsack CRS.