带时间窗口和延迟的缓存
Caching with Time Windows and Delays
摘要：
我们考虑两个概括的经典加权分页问题，将延迟服务的页面请求的概念。第一个是（加权）分页与时间窗口（PageTW）的问题，这是像经典的加权分页问题，除了每个页面请求只需要在给定的截止日期之前提供服务。这个问题出现在许多在线缓存的实际应用中，例如Linux内核中的"截止日期" I/O调度程序和视频点播流。第二个，也是更一般的问题是（加权）延迟寻呼（PageD）问题，其中服务页面请求的延迟导致对目标的惩罚。这个问题概括了缓存问题，以允许延迟服务，这是最近在在线算法中获得牵引力的一系列工作（例如，Emek等人。STOC '16，Azar等人。STOC' 17，Azar和Touitou FOCS '19）。我们给出了$O(\log k\log n)$-竞争算法的PageTW和PageD问题的$n$页与缓存大小$k$。这显著地改进了$O(k)$对于这两个问题的先前最佳界限（Azar等人，STOC '17）。我们还考虑了离线PageTW和PageD问题，给出了$O(1)$近似算法，并证明了APX-硬度。这些是离线问题的第一个结果;在我们的工作之前，甚至NP-硬度也不知道。在我们的算法的核心是一个新的"命中集" LP放松的PageTW问题，克服了$Ω(k)$完整性差距的自然LP的问题。据我们所知，这是第一个基于LP的算法用于具有延迟/截止日期的在线算法的例子。
英文摘要：
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 (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 (PageD) problem, where the delay in serving a page request results in a penalty being assessed 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., Emek et al. STOC '16, Azar et al. STOC '17, Azar and Touitou FOCS '19). We give $O(\log k\log n)$-competitive algorithms for both the PageTW and 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 (Azar et al. STOC '17). We also consider the offline PageTW and 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 PageTW problem that overcomes the $Ω(k)$ 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 algorithm with delays/deadlines.
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