New Results and Bounds on Online Facility Assignment Problem
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Consider an online facility assignment problem where a set of facilities F = { f_1, f_2, f_3, ⋯, 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 × c and customers arrive one by one having positions on the vertices of G. We show that Algorithm Greedy has competitive ratio r × c + r + c and Algorithm Optimal-Fill has competitive ratio O(r × 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.
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