On optimal matchings

@article{Ajtai1984OnOM,
  title={On optimal matchings},
  author={Mikl{\'o}s Ajtai and John Komlos and G{\'a}bor E. Tusn{\'a}dy},
  journal={Combinatorica},
  year={1984},
  volume={4},
  pages={259-264}
}
Givenn random red points on the unit square, the transportation cost between them is tipically √n logn. 
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214 Citations
Highly Influential Citations
38
Background Citations
82
Methods Citations
30
Results Citations
9

214 Citations

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Random restricted matching and lower bounds for combinatorial optimization
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This work gives a general framework for translating results from combinatorial optimization about the behaviour of random points into results for point sets with sufficiently high regularity and introduces a new irregularity problem concerning Voronoi cells.
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An application to sensor network is derived to derive the optimal transportation cost to the power of the maximal random bicolored matching, which is related to the expected distance between two identical general random processes.
On the Travelling Salesperson Problem in Many Dimensions
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It is shown that the length Tm of the shortest tour through X1, …, Xm satisfies limm∞ E(Tm)/m1−1/d = β(d) for a certain number β( d) for some numerical constant K.
Optimal matching and deterministic sampling
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This research examines some extremal point matching problems, exploring the dependence of matching weight with partition cardinality in vertex-weighted bipartite graphs and considers the problem of subset selection, providing several deterministic algorithms for point selection that are as good as or better than random subset selection according to various criteria.
Approximation algorithms for the Euclidean bipartite TSP
Optimal Random Matchings on Trees and Applications
TLDR
Upper bounds on several sets for which showing reasonable matching results would previously have been intractable are proved, e.g., the Cantor set, and various fractals.
Large and moderate deviations for matching problems and empirical discrepancies
We study the two-sample matching problem and its connections with the Monge-Kantorovich problem of optimal transportation of mass. We exploit this connection to obtain moderate and large deviation…
Convergence of asymptotic costs for random Euclidean matching problems
TLDR
The average minimum cost of a bipartite matching between two samples of n independent random points uniformly distributed on a unit cube in d ≥ 3 dimensions is investigated, where the matching cost is given by any power p ≥ 1 of their Euclidean distance.
Filling random cycles
  • Fedor Manin
  • Mathematics
    Commentarii Mathematici Helvetici
  • 2021
We compute the asymptotic behavior of the average-case filling volume for certain models of random Lipschitz cycles in the unit cube and sphere. For example, we estimate the minimal area of a Seifert…
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