Universal Optimization for Non-Clairvoyant Subadditive Joint Replenishment

TL;DR

This work develops a simple, modular non-clairvoyant framework for Subadditive JRP that matches the best known $O(\sqrt{n \log n})$ competitive ratio and delivers $O(\sqrt{n})$-competitive algorithms for Multi-Level Aggregation and Weighted Symmetric Subadditive JRP. The key technique is reducing any subadditive service function to a disjoint service function via a two-step approach: first model the approximation with Disjoint TCP Acknowledgement, then leverage Universal Set Cover (USC) to obtain such an approximation with a controlled stretch. Central to the results is the Reduction Lemma, which shows a 2$\alpha$-competitive non-clairvoyant algorithm for Subadditive JRP whenever a disjoint $\alpha$-approximation exists; using USC yields $O(\sqrt{n \log n})$, while structure in MLA and weighted symmetric cases yields $O(\sqrt{n})$. The framework thus unifies reductions to well-studied problems and establishes (i) tight $O(\sqrt{n})$ results for MLA and Weighted Symmetric JRP and (ii) matching lower bounds that certify the limits of current disjoint-approximation techniques. This work advances the design of robust non-clairvoyant online policies for a broad class of delay-aware replenishment problems with subadditive costs and highlights open questions about potentially beating the $O(\sqrt{n \log n})$ barrier for general subadditive costs.

Abstract

The online joint replenishment problem (JRP) is a fundamental problem in the area of online problems with delay. Over the last decade, several works have studied generalizations of JRP with different cost functions for servicing requests. Most prior works on JRP and its generalizations have focused on the clairvoyant setting. Recently, Touitou [Tou23a] developed a non-clairvoyant framework that provided an $O(\sqrt{n \log n})$ upper bound for a wide class of generalized JRP, where $n$ is the number of request types. We advance the study of non-clairvoyant algorithms by providing a simpler, modular framework that matches the competitive ratio established by Touitou for the same class of generalized JRP. Our key insight is to leverage universal algorithms for Set Cover to approximate arbitrary monotone subadditive functions using a simple class of functions termed \textit{disjoint}. This allows us to reduce the problem to several independent instances of the TCP Acknowledgement problem, for which a simple 2-competitive non-clairvoyant algorithm is known. The modularity of our framework is a major advantage as it allows us to tailor the reduction to specific problems and obtain better competitive ratios. In particular, we obtain tight $O(\sqrt{n})$-competitive algorithms for two significant problems: Multi-Level Aggregation and Weighted Symmetric Subadditive Joint Replenishment. We also show that, in contrast, Touitou's algorithm is $Ω(\sqrt{n \log n})$-competitive for both of these problems.

Figures (1)

• Figure 1: Tight instance for Touitou23, where $w = \frac{\sqrt{n \log n} - 1}{n-1}$.

Theorems & Definitions (13)

• Lemma 1.0: Reduction Lemma
• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.1: Reduction Lemma
• proof
• Lemma 2.2
• proof
• Theorem 2.3