PERMUTATION Strikes Back: The Power of Recourse in Online Metric Matching
Varun Gupta, Ravishankar Krishnaswamy, Sai Sandeep

TL;DR
This paper demonstrates that allowing a small amount of recourse in online metric matching significantly improves competitive ratios, achieving near-optimal solutions in general and line metrics with minimal adjustments.
Contribution
It introduces the first deterministic algorithms with recourse that surpass classical lower bounds for online metric matching in general and line metrics.
Findings
Deterministic $O( ext{log }k)$-competitive algorithm with $O( ext{log }k)$ recourse for general metrics.
Deterministic 3-competitive algorithm with $O( ext{log }k)$ recourse for line metrics.
Recourse greatly enhances the quality of online matchings under minimal modifications.
Abstract
In the classical Online Metric Matching problem, we are given a metric space with servers. A collection of clients arrive in an online fashion, and upon arrival, a client should irrevocably be matched to an as-yet-unmatched server. The goal is to find an online matching which minimizes the total cost, i.e., the sum of distances between each client and the server it is matched to. We know deterministic algorithms~\cite{KP93,khuller1994line} that achieve a competitive ratio of , and this bound is tight for deterministic algorithms. The problem has also long been considered in specialized metrics such as the line metric or metrics of bounded doubling dimension, with the current best result on a line metric being a deterministic competitive algorithm~\cite{raghvendra2018optimal}. Obtaining (or refuting) -competitive algorithms in general metrics and…
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