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… 
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References

SHOWING 1-10 OF 15 REFERENCES
The Online Metric Matching Problem for Doubling Metrics
TLDR
O(logk)-competitive randomized algorithms are given, which reduces the gap between the current O(log2k)- competitive randomized algorithms and the constant-competitive lower bounds known for these settings.
A Randomized O(log2k)-Competitive Algorithm for Metric Bipartite Matching
TLDR
An O(log2k)-competitive randomized algorithm is given for the online metric matching problem, which improves upon the best known guarantee of O( log3k) on the competitive factor due to Meyerson, Nanavati and Poplawski.
A o(n) -Competitive Deterministic Algorithm for Online Matching on a Line
TLDR
It is shown that online matching on a line is essentially equivalent to a particular search problem, that is, \(k\) -lost cows, and the first deterministic sub-linearly competitive algorithm is obtained by giving such an algorithm for the \(k\)-lost cows problem.
The Online Matching Problem on a Line
TLDR
This work shows that the generalized Work Function Algorithm has constant competitive ratio for the online matching problem, and shows that it is in fact Ω(logn) and O(n), and makes some progress towards proving a better upper bound by establishing some structural properties of the solutions.
A Robust and Optimal Online Algorithm for Minimum Metric Bipartite Matching
TLDR
This work presents a deterministic online algorithm that is the first to simultaneously achieve optimal performances in the well-known adversarial and the random arrival models and proves that any online algorithm will have a competitive ratio of at least 2H_n - 1-o(1) in this model.
Online Matching On a Line
An Input Sensitive Online Algorithm for the Metric Bipartite Matching Problem
  • K. Nayyar, S. Raghvendra
  • Computer Science, Mathematics
    2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2017
TLDR
A novel input sensitive analysis of a deterministic online algorithm for the minimum metric bipartite matching problem and shows that the cost of edges of the optimal matching inside each larger ball can be shown to be proportional to the weight times the radius of the larger ball.
SIGACT News Online Algorithms Column 27: Online Matching on the Line, Part 1
TLDR
This column contains a proof of a linear upper Bound for the generalized work function algorithm and a logarithmic lower bound for the algorithm, and conjecture that this algorithm in fact has a logrithmic competitive ratio (which would match the known lower bound), but this remains an open question.
SIGACT News Online Algorithms Column 28: Online Matching on the Line, Part 2
TLDR
This column gives a more detailed analysis of this algorithm, leading to a different (but again linear) upper bound, and conjecture that this algorithm in fact has a logarithmic competitive ratio (which would match the known lower bound for it), but this very much remains an open question.
The Online Transportation Problem
TLDR
The online transportation problem under the weakened adversary model is studied and an algorithm Balance is presented, which is a simple modification of the greedy algorithm that has a halfopt-competitive ratio of O(1).
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