Deterministic Min-Cost Matching with Delays

TL;DR

This paper presents the first deterministic algorithms for online minimum-cost matching with delays that achieve sub-linear competitive ratios, improving upon previous randomized methods.

Contribution

It introduces deterministic algorithms with sub-linear competitive ratios for online matching with delays, applicable to general metric spaces without prior knowledge.

Findings

Achieved $O(m^{0.59})$ competitive ratio for small $\\epsilon$

First deterministic algorithms with sub-linear competitive ratios for these problems

Applicable to general metric spaces without prior knowledge

Abstract

We consider the online Minimum-Cost Perfect Matching with Delays (MPMD) problem introduced by Emek et al. (STOC 2016), in which a general metric space is given, and requests are submitted in different times in this space by an adversary. The goal is to match requests, while minimizing the sum of distances between matched pairs in addition to the time intervals passed from the moment each request appeared until it is matched. In the online Minimum-Cost Bipartite Perfect Matching with Delays (MBPMD) problem introduced by Ashlagi et al. (APPROX/RANDOM 2017), each request is also associated with one of two classes, and requests can only be matched with requests of the other class. Previous algorithms for the problems mentioned above, include randomized $O\left(\log n\right)$-competitive algorithms for known and finite metric spaces, $n$ being the size of the metric space, and a deterministic $O\left(m\right)$-competitive algorithm, $m$ being the number of requests. We introduce $O\left(m^{\log\left(\frac{3}{2}+\epsilon\right)}\right)$-competitive deterministic algorithms for both problems and for any fixed $\epsilon > 0$. In particular, for a small enough $\epsilon$ the competitive ratio becomes $O\left(m^{0.59}\right)$. These are the first deterministic algorithms for the mentioned online matching problems, achieving a sub-linear competitive ratio. Our algorithms do not need to know the metric space in advance.