Mathematical Programming

, Volume 1, Issue 1, pp 6–25

The traveling-salesman problem and minimum spanning trees: Part II

  • Michael Held
  • Richard M. Karp

DOI: 10.1007/BF01584070

Cite this article as:
Held, M. & Karp, R.M. Mathematical Programming (1971) 1: 6. doi:10.1007/BF01584070


The relationship between the symmetric traveling-salesman problem and the minimum spanning tree problem yields a sharp lower bound on the cost of an optimum tour. An efficient iterative method for approximating this bound closely from below is presented. A branch-and-bound procedure based upon these considerations has easily produced proven optimum solutions to all traveling-salesman problems presented to it, ranging in size up to sixty-four cities. The bounds used are so sharp that the search trees are minuscule compared to those normally encountered in combinatorial problems of this type.


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Copyright information

© North-Holland Publishing Company 1971

Authors and Affiliations

  • Michael Held
    • 1
  • Richard M. Karp
    • 2
  1. 1.IBM Systems Research Institute New YorkNew YorkUSA
  2. 2.University of CaliforniaBerkeleyUSA

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