Online facility assignment

Abstract

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 ci to facility fj before the next customer ci+1 arrives. The cost of this assignment is the distance between ci and fj. 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.

Competitive Analysis for Two Variants of Online Metric Matching Problem

“…for OFAL(4) do not contradict the abovementioned upper bound of Optimal-fill, since upper bounds by Ahmed et al [1] are with respect to the asymptotic competitive ratio, while our lower bounds are with respect to the strict competitive ratio (see Sec. 2.3).…”

“…They studied the online matching problem on unweighted bipartite graphs with 2n vertices that contain a perfect matching, where the goal is to maximize the size of the obtained matching. In [9], they first showed that a deterministic greedy algorithm is 1 2 -competitive and optimal. They also presented a randomized algorithm Ranking and showed that it is (1− 1 e )-competitive and optimal.…”

“…Besides the problem on a line, Ahmed, Rahman and Kobourov [1] studied the online facility assignment problem on an unweighted graph G(V, E). They showed that the greedy algorithm is 2|E|-competitive and Optimal-Fill is |E|k rcompetitive, where |E| is the number of edge of G and r is the radius of G.…”

“…In 2020, Ahmed, Rahman and Kobourov [1] proposed a problem called the online facility assignment problem and considered it on a line, which we denote OFAL for short. In this problem, all the servers (which they call facilities) and requests (which they call customers) lie on a 1-dimensional line, and the distance between every pair of adjacent servers is the same.…”

“…In their model, all the servers are assumed to have the same capacity. Let us denote OFAL(k) the OFAL problem where the number of servers is k. Ahmed et al [1] showed that for OFAL(k) the greedy algorithm is 4k-competitive for any k and a deterministic algorithm Optimal-fill is k-competitive for any k > 2.…”