Consider an online facility assignment problem where a set of facilities $F = \{ f_1, f_2, f_3, \cdots, f_{|F|} \}$ of equal capacity $l$ is situated on a metric space and customers arrive one by one in an online manner on that space. We assign a customer $c_i$ to a facility $f_j$ before a new customer $c_{i+1}$ arrives. The cost of this assignment is the distance between $c_i$ and $f_j$. The objective of this problem is to minimize the sum of all assignment costs. Recently Ahmed et al. (TCS, 806, pp. 455-467, 2020) studied the problem where the facilities are situated on a line and computed competitive ratio of "Algorithm Greedy" which assigns the customer to the nearest available facility. They computed competitive ratio of algorithm named "Algorithm Optimal-Fill" which assigns the new customer considering optimal assignment of all previous customers. They also studied the problem where the facilities are situated on a connected unweighted graph. In this paper we first consider that $F$ is situated on the vertices of a connected unweighted grid graph $G$ of size $r \times c$ and customers arrive one by one having positions on the vertices of $G$. We show that Algorithm Greedy has competitive ratio $r \times c + r + c$ and Algorithm Optimal-Fill has competitive ratio $O(r \times c)$. We later show that the competitive ratio of Algorithm Optimal-Fill is $2|F|$ for any arbitrary graph. Our bound is tight and better than the previous result. We also consider the facilities are distributed arbitrarily on a plane and provide an algorithm for the scenario. We also provide an algorithm that has competitive ratio $(2n-1)$. Finally, we consider a straight line metric space and show that no algorithm for the online facility assignment problem has competitive ratio less than $9.001$.

中文翻译：

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在线设施分配问题的新结果和界限

考虑一个在线设施分配问题，其中一组设施 $F = \{ f_1, f_2, f_3, \cdots, f_{|F|} \}$ 等容量 $l$ 位于度量空间上，客户到达一个在该空间上以在线方式逐一。我们在新客户 $c_{i+1}$ 到达之前将客户 $c_i$ 分配给设施 $f_j$。此分配的成本是 $c_i$ 和 $f_j$ 之间的距离。这个问题的目标是最小化所有分配成本的总和。最近艾哈迈德等人。(TCS, 806, pp. 455-467, 2020) 研究了设施位于一条线上的问题，并计算了“算法贪婪”的竞争比率，将客户分配到最近的可用设施。他们计算了名为“Algorithm Optimal-Fill”的算法的竞争率 考虑所有以前客户的最佳分配来分配新客户。他们还研究了设施位于连通的未加权图上的问题。在本文中，我们首先考虑 $F$ 位于大小为 $r \times c$ 的连通未加权网格图 $G$ 的顶点上，并且客户一个接一个到达，在 $G$ 的顶点上有位置。我们表明算法贪婪具有竞争比率 $r \times c + r + c$ 和算法最优填充具有竞争比率 $O(r \times c)$。我们稍后证明，对于任意图，算法最优填充的竞争比率是 $2|F|$。我们的界限很紧，并且比之前的结果更好。我们还考虑设施在平面上任意分布，并为场景提供算法。我们还提供了一种具有竞争比率 $(2n-1)$ 的算法。最后，我们考虑了一个直线度量空间，并表明在线设施分配问题的算法没有低于 $9.001$ 的竞争率。