New Results and Bounds on Online Facility Assignment Problem

TL;DR

This paper investigates online facility assignment on various metric spaces, providing new bounds on algorithm competitiveness, including tight bounds for grid graphs, arbitrary graphs, and lower bounds for line metrics.

Contribution

It introduces improved competitive ratio bounds for greedy and optimal-fill algorithms on grid and arbitrary graphs, and establishes a lower bound for line metrics.

Findings

Algorithm Greedy has competitive ratio r*c + r + c on grid graphs.

Algorithm Optimal-Fill has competitive ratio O(r*c) on grid graphs.

No online algorithm can have a competitive ratio less than 9.001 on line metrics.

Abstract

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 $9.001$.