Set Cover with Delay – Clairvoyance is not Required
arXiv: Data Structures and Algorithms(2018)
摘要
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 Ω(√(log n)) and Ω(√(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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