Online Rounding for Set Cover under Subset Arrivals

TL;DR

This paper introduces an improved online rounding scheme for set cover problems under subset arrivals, achieving better approximation guarantees and extending to special cases like edge cover.

Contribution

It presents an O(log^2 s)-competitive rounding scheme for subset arrival models, improving previous bounds, and offers a 1.8-competitive scheme for edge cover.

Findings

Achieves O(log^2 s)-competitive ratio under subset arrival model.

Provides an improved approximation for multi-stage stochastic set cover.

Develops a 1.8-competitive scheme for edge cover.

Abstract

A rounding scheme for set cover has served as an important component in design of approximation algorithms for the problem, and there exists an H_s-approximate rounding scheme, where s denotes the maximum subset size, directly implying an approximation algorithm with the same approximation guarantee. A rounding scheme has also been considered under some online models, and in particular, under the element arrival model used as a crucial subroutine in algorithms for online set cover, an O(log s)-competitive rounding scheme is known [Buchbinder, Chen, and Naor, SODA 2014]. On the other hand, under a more general model, called the subset arrival model, only a simple O(log n)-competitive rounding scheme is known, where n denotes the number of elements in the ground set. In this paper, we present an O(log^2 s)-competitive rounding scheme under the subset arrival model, with one mild assumption that s is known upfront. Using our rounding scheme, we immediately obtain an O(log^2 s)-approximation algorithm for multi-stage stochastic set cover, improving upon the existing algorithms [Swamy and Shmoys, SICOMP 2012; Byrka and Srinivasan, SIDMA 2018] when s is small enough compared to the number of stages and the number of elements. Lastly, for set cover with s = 2, also known as edge cover, we present a 1.8-competitive rounding scheme under the edge arrival model.