Online Deterministic Minimum Cost Bipartite Matching with Delays on a Line

TL;DR

This paper presents a deterministic algorithm for online bipartite matching with delays on a line, improving the competitive ratio to approximately O(m^{0.5}) and leveraging the Robust Matching approach.

Contribution

It introduces a new delay-based matching algorithm for the line metric, achieving a better competitive ratio than previous methods for this specific case.

Findings

Achieves a deterministic old;O(m^{0.5})old; competitive ratio on the line.

Improves upon prior results for the line metric case.

Uses delay strategies based on the t-net-cost to optimize matching timing.

Abstract

We study the online minimum cost bipartite perfect matching with delays problem. In this problem, $m$ servers and $m$ requests arrive over time, and an online algorithm can delay the matching between servers and requests by paying the delay cost. The objective is to minimize the total distance and delay cost. When servers and requests lie in a known metric space, there is a randomized $O(\log n)$-competitive algorithm, where $n$ is the size of the metric space. When the metric space is unknown a priori, Azar and Jacob-Fanani proposed a deterministic $O\left(\frac{1}{\epsilon}m^{\log\left(\frac{3+\epsilon}{2}\right)}\right)$-competitive algorithm for any fixed $\epsilon > 0$. This competitive ratio is tight when $n = 1$ and becomes $O(m^{0.59})$ for sufficiently small $\epsilon$. In this paper, we improve upon the result of Azar and Jacob-Fanani for the case where servers and requests are on the real line, providing a deterministic $\tilde{O}(m^{0.5})$-competitive algorithm. Our algorithm is based on the Robust Matching (RM) algorithm proposed by Raghvendra for the minimum cost bipartite perfect matching problem. In this problem, delay is not allowed, and all servers arrive in the beginning. When a request arrives, the RM algorithm immediately matches the request to a free server based on the request's minimum $t$-net-cost augmenting path, where $t > 1$ is a constant. In our algorithm, we delay the matching of a request until its waiting time exceeds its minimum $t$-net-cost divided by $t$.