We study the \emph{Online Facility Assignment} (OFA) problem on a discrete $r\times c$ grid graph under the standard model of Ahmed, Rahman, and Kobourov: a fixed set of facilities is given, each with limited capacity, and an online sequence of unit-demand requests must be irrevocably assigned upon arrival to an available facility, incurring Manhattan ($L_1$) distance cost. We investigate how the discrete geometry of grids interacts with capacity depletion by analyzing two natural baselines and one capacity-aware heuristic. First, we give explicit adversarial sequences on grid instances showing that purely local rules can be forced into large competitive ratios: (i) a capacity-sensitive weighted-Voronoi heuristic (\textsc{CS-Voronoi}) can suffer cascading \emph{region-collapse} effects when nearby capacity is exhausted; and (ii) nearest-available \textsc{Greedy} (with randomized tie-breaking) can be driven into repeated long reassignments via an \emph{oscillation} construction. These results formalize geometric failure modes that are specific to discrete $L_1$ metrics with hard capacities. Motivated by these lower bounds, we then discuss a semi-online extension in which the algorithm may delay assignment for up to $τ$ time steps and solve each batch optimally via a min-cost flow computation. We present this batching framework as a remediation strategy and delineate the parameters that govern its performance, while leaving sharp competitive guarantees for this semi-online variant as an open direction.

翻译：我们在Ahmed、Rahman和Kobourov的标准模型下，研究离散$r\times c$网格图上的\emph{在线设施分配}（OFA）问题：给定一组固定的设施，每个设施具有有限容量，一个在线到达的单位需求请求序列必须在到达时被不可撤销地分配至一个可用设施，并产生曼哈顿（$L_1$）距离成本。通过分析两种自然基准算法和一种容量感知启发式算法，我们研究了网格的离散几何特性如何与容量耗尽相互作用。首先，我们在网格实例上给出显式的对抗序列，表明纯局部规则可能被迫产生较大的竞争比：（i）容量敏感的加权Voronoi启发式算法（\textsc{CS-Voronoi}）在附近容量耗尽时可能遭受级联的\emph{区域塌缩}效应；（ii）通过\emph{振荡}构造，最近可用\textsc{贪婪}算法（带随机平局打破）可能被驱动至重复的长距离重新分配。这些结果形式化了硬容量离散$L_1$度量所特有的几何失效模式。受这些下界启发，我们随后讨论一种半在线扩展，其中算法可将分配延迟最多$τ$个时间步，并通过最小成本流计算对每个批次进行最优求解。我们提出这种批处理框架作为一种补救策略，并界定了决定其性能的参数，同时将这种半在线变体的精确竞争性保证留作一个开放研究方向。