Set Cover with Delay -- Clairvoyance is not Required

TL;DR

This paper introduces the first non-clairvoyant algorithm for set cover with delay, achieving near-optimal competitiveness independent of request count, and establishes lower bounds showing clairvoyance offers limited advantage.

Contribution

It presents the first non-clairvoyant algorithm for set cover with delay with competitive ratio $O( ext{log } n ext{ log } m)$, matching the best clairvoyant results and providing new lower bounds.

Findings

Non-clairvoyant algorithm is $O( ext{log } n ext{ log } m)$-competitive.

Clairvoyance does not significantly improve competitiveness, with lower bounds of $ ext{Omega}( ext{sqrt(log n})$ and $ ext{Omega}( ext{sqrt(log m})$.

Algorithm's competitiveness is independent of the number of requests.

Abstract

In most online problems with delay, clairvoyance (i.e. knowing the future delay of a request upon its arrival) is required for polylogarithmic competitiveness. In this paper, we show that this is not the case for set cover with delay (SCD) -- specifically, we present the first non-clairvoyant algorithm, which is $O(\log n \log m)$-competitive, where $n$ is the number of elements and $m$ is the number of sets. This matches the best known result for the classic online set cover (a special case of non-clairvoyant SCD). Moreover, clairvoyance does not allow for significant improvement - we present lower bounds of $\Omega(\sqrt{\log n})$ and $\Omega(\sqrt{\log m})$ for SCD which apply for the clairvoyant case. In addition, the competitiveness of our algorithm does not depend on the number of requests. Such a guarantee on the size of the universe alone was not previously known even for the clairvoyant case - the only previously-known algorithm (due to Carrasco et al.) is clairvoyant, with competitiveness that grows with the number of requests. For the special case of vertex cover with delay, we show a simpler, deterministic algorithm which is $3$-competitive (and also non-clairvoyant).