Competitive Analysis for Two Variants of Online Metric Matching Problem

TL;DR

This paper analyzes two variants of the online metric matching problem, providing competitive ratio bounds for greedy algorithms and establishing lower bounds for server capacities on a line.

Contribution

It introduces bounds for the competitive ratios of algorithms in two online metric matching variants, including a greedy algorithm analysis and capacity-based lower bounds.

Findings

Greedy algorithm achieves a competitive ratio of 3 for the first problem.

Lower bounds on competitive ratios are established for server capacities 3, 4, and 5.

Matching lower bounds demonstrate the optimality of certain algorithms under given conditions.

Abstract

In this paper, we study two variants of the online metric matching problem. The first problem is 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 and give a matching lower bound. The second problem is the online facility assignment problem on a line, where servers have capacities, servers and requests are placed on 1-dimensional line, and the distances between any two consecutive servers are the same. We show lower bounds $1+ \sqrt{6}$ $(> 3.44948)$, $\frac{4+\sqrt{73}}{3}$ $(>4.18133)$ and $\frac{13}{3}$ $(>4.33333)$ on the competitive ratio when the numbers of servers are 3, 4 and 5, respectively.