# A Collection of Lower Bounds for Online Matching on the Line

@article{Antoniadis2018ACO,
title={A Collection of Lower Bounds for Online Matching on the Line},
author={Antonios Foivos Antoniadis and Carsten Fischer and Andreas T{\"o}nnis},
journal={ArXiv},
year={2018},
volume={abs/1712.07099}
}
• Published 19 December 2017
• Computer Science
• ArXiv
In the online matching on the line problem, the task is to match a set of requests $R$ online to a given set of servers $S$. The distance metric between any two points in $R\,\cup\, S$ is a line metric and the objective for the online algorithm is to minimize the sum of distances between matched server-request pairs. This problem is well-studied and - despite recent improvements - there is still a large gap between the best known lower and upper bounds: The best known deterministic algorithm…
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ArXiv
• 2020
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• Computer Science
ArXiv
• 2020
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APPROX-RANDOM
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This paper investigates the robustness of lower bounds of deterministic algorithms in general metrics by considering the Online Metric Matching with Recourse problem, and shows that a small logarithmic amount of recourse can significantly improve the quality of matchings the authors can maintain.
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• 2020
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• Computer Science
Algorithmica
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It is shown that online matching on a line is essentially equivalent to a particular search problem, which is called k-lost-cows, and the first deterministic sub-linearly competitive algorithm is obtained by giving such an algorithm for the k- Lost-Cows problem.
Capacity-Insensitive Algorithms for Online Facility Assignment Problems on a Line
• Computer Science
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• 2022
A class of algorithms called MPFS (most preferred free servers) is introduced and it is shown that any MPFS algorithm has the capacity-insensitive property, i.e., for any ℓ ≥ 1, alg is c -competitive for OFA( k, 1) iﬀ alg, and idas, the best possible in all the MPFS algorithms.
Matching on the Line Admits No o(√log n)-Competitive Algorithm
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ICALP
• 2021
We present a simple proof that the competitive ratio of any randomized online matching algorithm for the line exceeds √ log2(n+1)/15 for all n = 2−1 : i ∈ N, settling a 25-year-old open question.
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An error-dependent competitive ratio is given, which is a function of a user-de ned con dence parameter, and which interpolates smoothly between an optimal consistency, the performance in case that all predictions are correct, and the best-possible robustness regardless of the prediction quality.
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A poly-log competitive posted-price algorithm for online metrical searching in a tree metric inspired by demand-responsive parking pricing systems is presented.

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