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Random restricted matching and lower bounds for combinatorial optimization

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Abstract

We prove results on optimal random extensions of trees over points in [0,1]d. As an application, we give a general framework for translating results from combinatorial optimization about the behaviour of random points into results for point sets with sufficiently high regularity. We furthermore introduce a new irregularity problem concerning Voronoi cells, which has applications in logistics.

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References

  • Ajtai M, Komlós J, Tusnády G (1984) On optimal matchings. Combinatorica 4(4):259–264

    Article  MathSciNet  MATH  Google Scholar 

  • Anderssen R, Brent R, Daley D, Moran P (1976) Concerning \(\int_{0}^{1}{ \cdots\int_{0}^{1}{(x_{1}^{2}+\cdots x_{n}^{2})^{\frac{1}{2}}dx_{1}\cdots dx_{n}}}\) and a Taylor series method. SIAM J Appl Math 30:22–30

    Article  MathSciNet  MATH  Google Scholar 

  • Arnold B, Balakrishnan N, Nagaraja HN (2008) A first course in order statistics. Society for Industrial Mathematics

  • Bailey D, Borwein J, Crandall R (2007) Box integrals. J Comput Appl Math 206:196–208

    Article  MathSciNet  MATH  Google Scholar 

  • Bailey D, Borwein J, Crandall R (2010) Advances in the theory of box integrals. Math Comput 79(271):1830–1866

    MathSciNet  Google Scholar 

  • Beardwood J, Halton J, Hammersley J (1959) The shortest path through many points. Proc Camb Philos Soc 55:299–327

    Article  MathSciNet  MATH  Google Scholar 

  • Beck J (1989) A two-dimensional van Aardenne-Ehrenfest theorem in irregularities of distribution. Compos Math 72:269–339

    MATH  Google Scholar 

  • Drmota M, Tichy R (1997) Sequences, discrepancies and applications. In: Lecture notes in mathematics, vol 1651. Springer, Berlin

    Google Scholar 

  • Kuipers L, Niederreiter H (1974) Uniform distribution of sequences. Wiley, New York

    MATH  Google Scholar 

  • Liardet P (1979) Discrépancies sur le cercle. Primaths. I, Univ Marseille

  • Myerson G (1992) Discrepancy and distance between sets. Indag Math 3:193–201

    Article  MathSciNet  MATH  Google Scholar 

  • Roth KF (1954) On irregularities of distribution. Mathematika 1:73–79

    Article  MathSciNet  MATH  Google Scholar 

  • Steele J (1980) Shortest paths through pseudo-random points in the d-cube. Proc Am Math Soc 80(1):130–134

    MathSciNet  MATH  Google Scholar 

  • Steele J (1981) Subadditive Euclidean functionals and non-linear growth in geometric probability. Ann Probab 9:365–376

    Article  MathSciNet  MATH  Google Scholar 

  • Steele J (1988) Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann Probab 16:1767–1787

    Article  MathSciNet  MATH  Google Scholar 

  • Steinerberger S (2010) A new lower bound for the geometric traveling salesman problem. Oper Res Lett 38(4):318–319

    Article  MathSciNet  MATH  Google Scholar 

  • Steinerberger S (2011) Extremal uniform distribution and random chord lengths. Acta Math Hung (accepted)

  • Talagrand M (1994) The transportation cost from the uniform measure to the empirical measure in dimension ≥3. Ann Probab 22(2):919–959

    Article  MathSciNet  MATH  Google Scholar 

  • van der Corput J, Pisot C (1939) Sur la discrépance modulo un. Indag Math 1:260–269

    Google Scholar 

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Authors and Affiliations

  1. Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Stefan Steinerberger

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  1. Stefan Steinerberger
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Correspondence to Stefan Steinerberger.

Additional information

This research was supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.

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Steinerberger, S. Random restricted matching and lower bounds for combinatorial optimization. J Comb Optim 24, 280–298 (2012). https://doi.org/10.1007/s10878-011-9384-4

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  • DOI: https://doi.org/10.1007/s10878-011-9384-4

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