A Learning Perspective on Random-Order Covering Problems

TL;DR

This paper addresses online covering problems under a random arrival order by establishing a direct, modular link between online convex optimization (OCO) and competitive analysis. By treating the learning component as a black-box OCO subroutine and feeding it problem-specific concave gain functions, the authors derive $O(\log mn)$-competitive algorithms for a range of random-order problems, including weighted set cover, set multicover, covering integer programs, and non-metric facility location. The core insight is that additive/multiplicative regret bounds for stochastic OCO translate into strong competitiveness for the online covering problems, yielding a unified framework that subsumes previous LearnOrCover results and extends to new settings. This framework provides a conceptually clean, modular approach with potential for broad applicability to monotone covering problems in the random-order model, potentially guiding future extensions to box-constrained and other complex variants.

Abstract

In the random-order online set cover problem, the instance with $m$ sets and $n$ elements is chosen in a worst-case fashion, but then the elements arrive in a uniformly random order. Can this random-order model allow us to circumvent the bound of $O(\log m \log n)$-competitiveness for the adversarial arrival order model? This long-standing question was recently resolved by Gupta et al. (2021), who gave an algorithm that achieved an $O(\log mn)$-competitive ratio. While their LearnOrCover was inspired by ideas in online learning (and specifically the multiplicative weights update method), the analysis proceeded by showing progress from first principles. In this work, we show a concrete connection between random-order set cover and stochastic mirror-descent/online convex optimization. In particular, we show how additive/multiplicative regret bounds for the latter translate into competitiveness for the former. Indeed, we give a clean recipe for this translation, allowing us to extend our results to covering integer programs, set multicover, and non-metric facility location in the random order model, matching (and giving simpler proofs of) the previous applications of the LearnOrCover framework.

Theorems & Definitions (27)

• Theorem
• Definition 2.1: OCO with Stochastic First-Order Feedback
• Theorem 2.2: OCO Algorithm
• Theorem 3.1
• Lemma 3.2: Bounding the Real Gain
• proof
• Lemma 3.3: Gain vs. Real Gain
• proof
• Lemma 3.4
• proof