Competitive analysis for two variants of online metric matching problem

Toshiya Itoh,Makoto Satake,Shuichi Miyazaki

doi:10.1142/s1793830921501561

Toshiya Itoh, Makoto Satake + Show 1 more

Open Access

https://doi.org/10.1142/s1793830921501561

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Abstract

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.

Highlights

The online metric matching problem was introduced independently by Kalyanasundaram and Pruhs [7] and Khuller, Mitchell and Vazirani [10]
We study a variant of the online metric matching problem where all the servers are placed at one of two positions in the metric space
We show that a simple greedy algorithm achieves the competitive ratio of 3 for this problem, and show that any deterministic online algorithm has competitive ratio at least

Summary

Introduction

The online metric matching problem was introduced independently by Kalyanasundaram and Pruhs [7] and Khuller, Mitchell and Vazirani [10]. In 1998, Kalyanasundaram and Pruhs [8] posed a question whether we can have a better competitive ratio by restricting the metric space to a line, and introduced the problem called the online matching problem on a line. Since this problem has been extensively studied, but there still remains a large gap between the best known lower bound 9.001 [5] and upper bound O(log n) [16]. Are with respect to the asymptotic competitive ratio, while our lower bounds are with respect to the strict competitive ratio (see Sec. 2.3)

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