A Learning Perspective on Random-Order Covering Problems

TL;DR

This paper establishes a connection between random-order set cover problems and online convex optimization, providing a unified framework to analyze competitiveness and extend results to related covering problems.

Contribution

It introduces a novel link between random-order set cover and stochastic mirror descent, simplifying analysis and enabling extensions to various covering problems.

Findings

Regret bounds translate into competitiveness guarantees.

Unified framework simplifies proofs for multiple covering problems.

Extends previous results to new problem 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.