Set Cover with Delay -- Clairvoyance is not Required

Yossi Azar; Ashish Chiplunkar; Shay Kutten; Noam Touitou

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arXiv · 1807.08543

Set Cover with Delay -- Clairvoyance is not Required

Yossi Azar·Ashish Chiplunkar·Shay Kutten·Noam Touitou

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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).

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Yossi Azar, Ashish Chiplunkar, Shay Kutten, Noam Touitou. 2018-07-23. Set Cover with Delay -- Clairvoyance is not Required. https://arxiv.org/abs/1807.08543

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