Online Facility Assignment

We consider the online facility assignment problem, with a set of facilities F of equal capacity l in metric space and customers arriving one by one in an online manner. We must assign customer \(c_i\) to facility \(f_j\) before the next customer \(c_{i+1}\) arrives. The cost of this assignment is the distance between \(c_i\) and \(f_j\). The total number of customers is at most |F|l and each customer must be assigned to a facility. The objective is to minimize the sum of all assignment costs. We first consider the case where facilities are placed on a line so that the distance between adjacent facilities is the same and customers appear anywhere on the line. We describe a greedy algorithm with competitive ratio 4|F| and another one with competitive ratio |F|. Finally, we consider a variant in which the facilities are placed on the vertices of a graph and two algorithms in that setting.