We consider the problem of online service with delay on a general metric space, first presented by Azar, Ganesh, Ge and Panigrahi (STOC 2017). The best known randomized algorithm for this problem, by Azar and Touitou (FOCS 2019), is $O(\log^2 n)$-competitive, where $n$ is the number of points in the metric space. This is also the best known result for the special case of online service with deadlines, which is of independent interest. In this paper, we present $O(\log n)$-competitive deterministic algorithms for online service with deadlines or delay, improving upon the results from FOCS 2019. Furthermore, our algorithms are the first deterministic algorithms for online service with deadlines or delay which apply to general metric spaces and have sub-polynomial competitiveness.

翻译：我们考虑一般度量空间中的延迟在线服务问题，该问题最早由Azar、Ganesh、Ge和Panigrahi（STOC 2017）提出。针对该问题，Azar与Touitou（FOCS 2019）给出的最优随机化算法达到$O(\log^2 n)$竞争力，其中$n$为度量空间中的点数。该结果同时也是具有独立研究价值的限时在线服务特例问题的最优已知结论。本文针对限时或延迟在线服务问题，提出了$O(\log n)$竞争力的确定性算法，改进了FOCS 2019的研究成果。此外，我们的算法是首个适用于一般度量空间且具有亚多项式竞争力的限时或延迟在线服务确定性算法。