This paper presents a new research direction for online Multi-Level Aggregation (MLA) with delays. In this problem, we are given an edge-weighted rooted tree $T$, and we have to serve a sequence of requests arriving at its vertices in an online manner. Each request $r$ is characterized by two parameters: its arrival time $t(r)$ and location $l(r)$ (a vertex). Once a request $r$ arrives, we can either serve it immediately or postpone this action until any time $t > t(r)$. We can serve several pending requests at the same time, and the service cost of a service corresponds to the weight of the subtree that contains all the requests served and the root of $T$. Postponing the service of a request $r$ to time $t > t(r)$ generates an additional delay cost of $t - t(r)$. The goal is to serve all requests in an online manner such that the total cost (i.e., the total sum of service and delay costs) is minimized. The current best algorithm for this problem achieves a competitive ratio of $O(d^2)$ (Azar and Touitou, FOCS'19), where $d$ denotes the depth of the tree. Here, we consider a stochastic version of MLA where the requests follow a Poisson arrival process. We present a deterministic online algorithm which achieves a constant ratio of expectations, meaning that the ratio between the expected costs of the solution generated by our algorithm and the optimal offline solution is bounded by a constant. Our algorithm is obtained by carefully combining two strategies. In the first one, we plan periodic oblivious visits to the subset of frequent vertices, whereas in the second one, we greedily serve the pending requests in the remaining vertices. This problem is complex enough to demonstrate a very rare phenomenon that ``single-minded" or ``sample-average" strategies are not enough in stochastic optimization.

翻译：本文提出了一种在线多级聚合（MLA）问题的新研究方向，该问题考虑了延迟因素。在此问题中，我们给定一棵边权有根树$T$，需要以在线方式处理其顶点上到达的请求序列。每个请求$r$由两个参数表征：到达时间$t(r)$和位置$l(r)$（一个顶点）。当请求$r$到达时，我们可以立即处理它，或将此操作推迟到任意时刻$t > t(r)$。我们可以同时处理多个待处理请求，服务的成本等于包含所有已处理请求及树$T$根的子树权重。将请求$r$的服务延迟到时刻$t > t(r)$会产生额外的延迟成本$t - t(r)$。目标是以在线方式处理所有请求，使得总成本（即服务成本与延迟成本之和）最小化。目前针对该问题的最优算法实现了$O(d^2)$的竞争比（Azar and Touitou, FOCS'19），其中$d$表示树的深度。本文考虑MLA的随机版本，其中请求遵循泊松到达过程。我们提出了一种确定性在线算法，该算法实现了期望值的常数比值，即我们算法生成的解的期望成本与最优离线解的期望成本之比有界于一个常数。我们的算法通过仔细结合两种策略得到。第一种策略是计划定期对高频顶点子集进行周期性访问；第二种策略是对剩余顶点中的待处理请求进行贪婪处理。该问题的复杂性足以展示一种非常罕见的现象：在随机优化中，“单一目标”或“样本平均”策略是不够的。