On the Value of a Random Minimum Weight Steiner Tree

Consider a complete graph on n vertices with edge weights chosen randomly and independently from an exponential distribution with parameter 1. Fix k vertices and consider the minimum weight Steiner tree which contains these vertices. We prove that with high probability the weight of this tree is (1+o(1))(k-1)(log n-log k)/n when k =o(n) and n→∞.

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Correspondence to Béla Bollobás*.

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* Research supported in part by NSF grant DSM9971788

† Research supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey. Part of this research was done while visiting IBM T. J. Watson Research Center.

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Bollobás*, B., Gamarnik, D., Riordan, O. et al. On the Value of a Random Minimum Weight Steiner Tree. Combinatorica 24, 187–207 (2004). https://doi.org/10.1007/s00493-004-0013-z

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Mathematics Subject Classification (2000):

  • 05C80
  • 60C05
  • 68R10