O(depth)-Competitive Algorithm for Online Multi-level Aggregation

We consider two generalizations of the classical weighted paging problem that incorporate the notion of delayed service of page requests. The first is the (weighted) paging with time windows (\sf PageTW) problem, which is like the classical weighted paging problem except that each page request only needs to be served before a given deadline. This problem arises in many practical applications of online caching, such as the “deadline” I/O scheduler in the Linux kernel and video-on-demand streaming. The second, and more general, problem is the (weighted) paging with delay (\sf PageD) problem, where the delay in serving a page request results in a penalty being added to the objective. This problem generalizes the caching problem to allow delayed service, a line of work that has recently gained traction in online algorithms (e.g., [Y. Emek, S. Kutten, and R. Wattenhofer, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, 2016, pp. 333--344; Y. Azar et al., Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, 2017, pp. 551--563; Y. Azar and N. Touitou, Proceedings of the 60th IEEE Annual Symposium on Foundations of Computer Science, 2019, pp. 60--71]). We give $O(\log k\log n)$-competitive algorithms for both the \sf PageTW and \sf PageD problems on $n$ pages with a cache of size $k$. This significantly improves on the previous best bounds of $O(k)$ for both problems [Y. Azar et al., Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, 2017, pp. 551--563]. We also consider the offline \sf PageTW and \sf PageD problems, for which we give $O(1)$-approximation algorithms and prove APX-hardness. These are the first results for the offline problems; even NP-hardness was not known before our work. At the heart of our algorithms is a novel “hitting-set” LP relaxation of the \sf PageTW problem that overcomes the $\Omega(k)$ integrality gap of the natural LP for the problem. To the best of our knowledge, this is the first example of an LP-based algorithm for an online problem with delays/deadlines.