# A Match in Time Saves Nine: Deterministic Online Matching With Delays

## TL;DR

This paper introduces the first deterministic online algorithm for Min-cost Perfect Matching with Delays, achieving a polynomial competitive ratio independent of metric parameters, advancing online matching theory.

## Contribution

It presents the first deterministic algorithm for MPMD with a polynomial competitive ratio, not requiring prior knowledge of the metric space.

## Key findings

- Deterministic algorithm with $O(m^{2.46})$ competitive ratio.
- Algorithm does not depend on metric space parameters.
- First such deterministic solution for MPMD.

## Abstract

We consider the problem of online Min-cost Perfect Matching with Delays (MPMD) introduced by Emek et al. (STOC 2016). In this problem, an even number of requests appear in a metric space at different times and the goal of an online algorithm is to match them in pairs. In contrast to traditional online matching problems, in MPMD all requests appear online and an algorithm can match any pair of requests, but such decision may be delayed (e.g., to find a better match). The cost is the sum of matching distances and the introduced delays. We present the first deterministic online algorithm for this problem. Its competitive ratio is $O(m^{\log_2 5.5})$ $ = O(m^{2.46})$, where $2 m$ is the number of requests. This is polynomial in the number of metric space points if all requests are given at different points. In particular, the bound does not depend on other parameters of the metric, such as its aspect ratio. Unlike previous (randomized) solutions for the MPMD problem, our algorithm does not need to know the metric space in advance.