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：本文提出了在线多级聚合（MLA）延迟问题的一个新研究方向。在该问题中，我们给定一棵带权有根树$T$，需要以在线方式处理一系列到达其顶点的请求。每个请求$r$由两个参数表征：到达时间$t(r)$和位置$l(r)$（即顶点）。请求$r$到达后，我们可以立即处理，或将处理动作推迟至任意时刻$t > t(r)$。我们可以同时处理多个待处理请求，单次处理的服务成本对应于包含所有被处理请求及树$T$根节点的子树权重。将请求$r$延迟至$t > t(r)$时刻处理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的总和）最小化。该问题当前最佳算法（Azar和Touitou，FOCS'19）的竞争比为$O(d^2)$，其中$d$表示树的深度。本文研究MLA的随机版本，其中请求服从泊松到达过程。我们提出一种确定性在线算法，该算法能实现期望值意义上的常数竞争比，即算法生成解的期望成本与最优离线解期望成本之比存在常数上界。我们的算法通过精心结合两种策略获得：第一种策略对高频顶点子集进行周期性无感知访问规划，第二种策略则贪婪地处理剩余顶点中的待处理请求。该问题的复杂性足以展现随机优化中“单一策略”或“样本平均策略”不足的罕见现象。