# 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} }

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…

## 12 Citations

Online Minimum Cost Matching on the Line with Recourse

- Computer ScienceArXiv
- 2020

This work gives a $(1+\varepsilon)-competitive algorithm that reassigns any request at most $O(n\log n)$ times, the first non-trivial result for min-cost bipartite matching with recourse and obtains a near-optimal result.

Matching on the line admits no $o(\sqrt{\log n})$-competitive algorithm

- Computer ScienceArXiv
- 2020

We present a simple proof that the competitive ratio of any randomized online matching algorithm for the line is at least √ log 2 (n+1)/12 for all n = 2−1 : i ∈ N. 1 Online matching, on the line In…

Online Minimum Cost Matching with Recourse on the Line

- Computer ScienceAPPROX-RANDOM
- 2020

This work shows an O(1)competitive algorithm for online matching on the line with amortized recourse of O(log n), the first non-trivial result for min-cost bipartite matching with recourse, and gives a (1+ε)-competitive algorithm that reassigns any request at most O(ε−1) times.

PERMUTATION Strikes Back: The Power of Recourse in Online Metric Matching

- Computer Science, MathematicsAPPROX-RANDOM
- 2020

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.

Competitive Analysis for Two Variants of Online Metric Matching Problem

- Mathematics, Computer ScienceCOCOA
- 2020

This paper purses competitive analysis for two variants of the online metric matching problem, one of which is the online facility assignment problem on a line and the other is a restriction where each server is placed at one of two positions.

A $${o}\mathopen {}\left( n\right) \mathclose {}$$on-Competitive Deterministic Algorithm for Online Matching on a Line

- Computer ScienceAlgorithmica
- 2019

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 ScienceArXiv
- 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

- Computer ScienceICALP
- 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.…

Double Coverage with Machine-Learned Advice

- Computer ScienceITCS
- 2022

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.

Competitively Pricing Parking in a Tree

- Computer Science, BusinessWINE
- 2020

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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