A randomized algorithm for the on-line weighted bipartite matching problem

TL;DR

This paper presents a randomized algorithm for the online weighted bipartite matching problem in metric spaces, achieving a sublogarithmic expected competitive ratio, improving upon deterministic bounds.

Contribution

It introduces a randomized algorithm that attains an expected competitive ratio of o(log^3 n), surpassing the deterministic ratio of Θ(n) for the problem.

Findings

Expected competitive ratio of o(log^3 n) for the algorithm.

Applicable to the fire station problem on the real line.

Improves bounds for online weighted bipartite matching.

Abstract

We study the on-line minimum weighted bipartite matching problem in arbitrary metric spaces. Here, $n$ not necessary disjoint points of a metric space $M$ are given, and are to be matched on-line with $n$ points of $M$ revealed one by one. The cost of a matching is the sum of the distances of the matched points, and the goal is to find or approximate its minimum. The competitive ratio of the deterministic problem is known to be $\Theta(n)$. It was conjectured that a randomized algorithm may perform better against an oblivious adversary, namely with an expected competitive ratio $\Theta(\log n)$. We prove a slightly weaker result by showing a $o(\log^3 n)$ upper bound on the expected competitive ratio. As an application the same upper bound holds for the notoriously hard fire station problem, where $M$ is the real line.