Competitive analysis for two variants of online metric matching problem

• Published in: Discrete mathematics, algorithms, and applications Vol. 13; no. 6
• Main Authors: Itoh, Toshiya, Miyazaki, Shuichi, Satake, Makoto
• Format: Journal Article
• Language: English
• Published: Singapore World Scientific Publishing Company 01.12.2021; World Scientific Publishing Co. Pte., Ltd
• Subjects: Algorithms; Competition; Greedy algorithms; Lower bounds; Matching; Metric space; Operations research; Ratios; online metric matching; Online algorithm; online facility assignment problem; competitive analysis
• ISSN: 1793-8309; 1793-8317
• DOI: 10.1142/S1793830921501561

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( 2 ). We show that a simple greedy algorithm achieves the competitive ratio of 3 for OMM( 2 ). We also show that this greedy algorithm is optimal by showing that the competitive ratio of any deterministic online algorithm for OMM( 2 ) 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( k ), where k is the number of servers. We first observe that the upper and lower bounds for OMM( 2 ) also hold for OFAL( 2 ), so the competitive ratio for OFAL( 2 ) is exactly 3. We then show lower bounds on the competitive ratio 1 + 6 ( > 3 . 4 4 9 4 8 ) , 4 + 7 3 3 ( > 4 . 1 8 1 3 3 ) and 1 3 3 ( > 4 . 3 3 3 3 3 ) for OFAL( 3 ), OFAL( 4 ) and OFAL( 5 ), respectively.