Journal of Combinatorial Optimization

, Volume 24, Issue 3, pp 280–298 | Cite as

Random restricted matching and lower bounds for combinatorial optimization

  • Stefan SteinerbergerEmail author


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.


Traveling salesman Minimal spanning tree Voronoi decomposition Restricted matching Packing problems Uniform distribution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ajtai M, Komlós J, Tusnády G (1984) On optimal matchings. Combinatorica 4(4):259–264 MathSciNetzbMATHCrossRefGoogle Scholar
  2. 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 MathSciNetzbMATHCrossRefGoogle Scholar
  3. Arnold B, Balakrishnan N, Nagaraja HN (2008) A first course in order statistics. Society for Industrial Mathematics Google Scholar
  4. Bailey D, Borwein J, Crandall R (2007) Box integrals. J Comput Appl Math 206:196–208 MathSciNetzbMATHCrossRefGoogle Scholar
  5. Bailey D, Borwein J, Crandall R (2010) Advances in the theory of box integrals. Math Comput 79(271):1830–1866 MathSciNetGoogle Scholar
  6. Beardwood J, Halton J, Hammersley J (1959) The shortest path through many points. Proc Camb Philos Soc 55:299–327 MathSciNetzbMATHCrossRefGoogle Scholar
  7. Beck J (1989) A two-dimensional van Aardenne-Ehrenfest theorem in irregularities of distribution. Compos Math 72:269–339 zbMATHGoogle Scholar
  8. Drmota M, Tichy R (1997) Sequences, discrepancies and applications. In: Lecture notes in mathematics, vol 1651. Springer, Berlin Google Scholar
  9. Kuipers L, Niederreiter H (1974) Uniform distribution of sequences. Wiley, New York zbMATHGoogle Scholar
  10. Liardet P (1979) Discrépancies sur le cercle. Primaths. I, Univ Marseille Google Scholar
  11. Myerson G (1992) Discrepancy and distance between sets. Indag Math 3:193–201 MathSciNetzbMATHCrossRefGoogle Scholar
  12. Roth KF (1954) On irregularities of distribution. Mathematika 1:73–79 MathSciNetzbMATHCrossRefGoogle Scholar
  13. Steele J (1980) Shortest paths through pseudo-random points in the d-cube. Proc Am Math Soc 80(1):130–134 MathSciNetzbMATHGoogle Scholar
  14. Steele J (1981) Subadditive Euclidean functionals and non-linear growth in geometric probability. Ann Probab 9:365–376 MathSciNetzbMATHCrossRefGoogle Scholar
  15. Steele J (1988) Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann Probab 16:1767–1787 MathSciNetzbMATHCrossRefGoogle Scholar
  16. Steinerberger S (2010) A new lower bound for the geometric traveling salesman problem. Oper Res Lett 38(4):318–319 MathSciNetzbMATHCrossRefGoogle Scholar
  17. Steinerberger S (2011) Extremal uniform distribution and random chord lengths. Acta Math Hung (accepted) Google Scholar
  18. Talagrand M (1994) The transportation cost from the uniform measure to the empirical measure in dimension ≥3. Ann Probab 22(2):919–959 MathSciNetzbMATHCrossRefGoogle Scholar
  19. van der Corput J, Pisot C (1939) Sur la discrépance modulo un. Indag Math 1:260–269 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of BonnBonnGermany

Personalised recommendations