Learning-Augmented Online Algorithms for Nonclairvoyant Joint Replenishment Problem with Deadlines

TL;DR

This work introduces learning-augmented online algorithms for the Joint Replenishment Problem with Deadlines (JRP-D) under deadline predictions. By formalizing an error metric based on instantaneous item inversions, the authors design a deterministic Local-Greedy algorithm that balances robustness and consistency, achieving a competitive ratio of $O\left(\min(\eta^{1/3}\log^{2/3} n, \sqrt{\eta}, \sqrt{n})\right)$ when the true deadline information is partially revealed. They further refine the analysis for uniform weights, bucket the items by cost, and demonstrate that these methods are combinable with a nonclairvoyant baseline to obtain a best-of-three guarantee. A matching lower bound within a natural restricted model shows near-optimality of the proposed approach for deterministic semi-memoryless algorithms. Overall, the paper charts a principled path to leveraging deadline predictions to close the gap between clairvoyant and nonclairvoyant online JRP-D, with clear psychophysics of prediction error via the $\eta$ metric and practical implications for robust scheduling under uncertainty.

Abstract

This paper considers using predictions in the context of the online Joint Replenishment Problem with Deadlines (JRP-D). Prior work includes asymptotically optimal competitive ratios of $O(1)$ for the clairvoyant setting and $O(\sqrt{n})$ of the nonclairvoyant setting, where $n$ is the number of items. The goal of this paper is to significantly reduce the competitive ratio for the nonclairvoyant case by leveraging predictions: when a request arrives, the true deadline of the request is not revealed, but the algorithm is given a predicted deadline. The main result is an algorithm whose competitive ratio is $O(\min(η^{1/3}\log^{2/3}(n), \sqrtη, \sqrt{n}))$, where $n$ is the number of item types and $η\leq n^2$ quantifies how flawed the predictions are in terms of the number of ``instantaneous item inversions.'' Thus, the algorithm is robust, i.e., it is never worse than the nonclairvoyant solution, and it is consistent, i.e., if the predictions exhibit no inversions, then the algorithm behaves similarly to the clairvoyant algorithm. Moreover, if the error is not too large, specifically $η< o(n^{3/2}/\log^2(n))$, then the algorithm obtains an asymptotically better competitive ratio than the nonclairvoyant algorithm. We also show that all deterministic algorithms falling in a certain reasonable class of algorithms have a competitive ratio of $Ω(η^{1/3})$, so this algorithm is nearly the best possible with respect to this error metric.

Theorems & Definitions (74)

• Theorem 1.1
• Theorem 1.2
• Lemma 1.3
• proof
• Theorem 1.4
• Definition 2.1: Request Inversion
• Definition 2.2: Item Inversion
• Definition 2.3: Instantaneous Inversions
• Proposition 3.1
• proof