Open AccessCompetitive analysis for two variants of online metric matching problem
Author(s) -
Toshiya Itoh,
Shuichi Miyazaki,
Makoto Satake
Publication year - 2021
Publication title -
discrete mathematics algorithms and applications/discrete mathematics, algorithms, and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.307
H-Index - 16
eISSN - 1793-8317
pISSN - 1793-8309
DOI - 10.1142/s1793830921501561
Subject(s) - competitive analysis , online algorithm , server , matching (statistics) , metric (unit) , combinatorics , metric space , upper and lower bounds , space (punctuation) , task (project management) , greedy algorithm , computer science , mathematics , discrete mathematics , algorithm , statistics , world wide web , economics , operating system , mathematical analysis , operations management , management
In the online metric matching problem, there are servers on a given metric space and requests are given one-by-one. The task of an online algorithm is to match each request immediately and irrevocably with one of the unused servers. In this paper, we pursue competitive analysis for two variants of the online metric matching problem. The first variant is a restriction where each server is placed at one of two positions, which is denoted by OMM([Formula: see text]). We show that a simple greedy algorithm achieves the competitive ratio of 3 for OMM([Formula: see text]). We also show that this greedy algorithm is optimal by showing that the competitive ratio of any deterministic online algorithm for OMM([Formula: see text]) is at least 3. The second variant is the online facility assignment problem on a line. In this problem, the metric space is a line, the servers have capacities, and the distances between any two consecutive servers are the same. We denote this problem by OFAL([Formula: see text]), where [Formula: see text] is the number of servers. We first observe that the upper and lower bounds for OMM([Formula: see text]) also hold for OFAL([Formula: see text]), so the competitive ratio for OFAL([Formula: see text]) is exactly 3. We then show lower bounds on the competitive ratio [Formula: see text] [Formula: see text], [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] for OFAL([Formula: see text]), OFAL([Formula: see text]) and OFAL([Formula: see text]), respectively.