# Stochastic Online Metric Matching

## TL;DR

This paper introduces improved algorithms for the stochastic online metric matching problem under i.i.d arrivals, achieving significantly better competitiveness than previous models, and demonstrates a clear separation from adversarial and random order models.

## Contribution

The paper presents an $O((\log \log \log n)^2)$-competitive algorithm for the i.i.d arrival model and a 9-competitive algorithm for line and tree metrics, advancing the understanding of stochastic online matching.

## Key findings

- Achieved $O((\log \log \log n)^2)$-competitiveness in the i.i.d model.
- Provided a 9-competitive algorithm for line and tree metrics.
- Established a separation between i.i.d, adversarial, and random order models.

## Abstract

We study the minimum-cost metric perfect matching problem under online i.i.d arrivals. We are given a fixed metric with a server at each of the points, and then requests arrive online, each drawn independently from a known probability distribution over the points. Each request has to be matched to a free server, with cost equal to the distance. The goal is to minimize the expected total cost of the matching. Such stochastic arrival models have been widely studied for the maximization variants of the online matching problem; however, the only known result for the minimization problem is a tight $O(\log n)$-competitiveness for the random-order arrival model. This is in contrast with the adversarial model, where an optimal competitive ratio of $O(\log n)$ has long been conjectured and remains a tantalizing open question. In this paper, we show improved results in the i.i.d arrival model. We show how the i.i.d model can be used to give substantially better algorithms: our main result is an $O((\log \log \log n)^2)$-competitive algorithm in this model. Along the way we give a $9$-competitive algorithm for the line and tree metrics. Both results imply a strict separation between the i.i.d model and the adversarial and random order models, both for general metrics and these much-studied metrics.