Mathematical Programming

, Volume 1, Issue 1, pp 6–25 | Cite as

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

  • Michael Held
  • Richard M. Karp


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.


Mathematical Method Iterative Method Span Tree Search Tree Minimum Span Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Agmon, “The relaxation method for linear inequalities,”Canadian Journal of Mathematics 6 (1954) 382–392.Google Scholar
  2. [2]
    M. Bellmore and G.L. Nemhauser, “The traveling salesman problem: a survey”,Operations Research 16 (1968) 538–558.Google Scholar
  3. [3]
    G.A. Croes, “A method for solving traveling salesman problems,”Operations Research 6 (1958) 791–812.Google Scholar
  4. [4]
    G.B. Dantzig, D.R. Fulkerson and S.M. Johnson, “Solution of a large scale traveling salesman problem”,Operation Research 2 (1954) 393–410.Google Scholar
  5. [5]
    E.W. Dijkstra, “A note on two problems in connexion with graphs,”Numerische Mathematik 1 (1959) 269–271.Google Scholar
  6. [6]
    D. Gale, “Optimal assignments in an ordered set: an application of matroid theory,”Journal of Combinatorial Theory 4 (1968) 176–180.Google Scholar
  7. [7]
    M. Held and R.M. Karp, “The traveling-salesman problem and minimum spanning trees,”Operations Research 18 (1970) 1138–1162.Google Scholar
  8. [8]
    M. Held and R.M. Karp, “A dynamic programming approach to sequencing problems,”Journal of the Society for Industrial and Applied Mathematics 10 (1962) 196–210.Google Scholar
  9. [9]
    L.L. Karg and G.L. Thompson, “A heuristic approach to solving traveling salesman problems,”Management Science 10 (1964) 225–248.Google Scholar
  10. [10]
    J.B. Kruskal, “On the shortest spanning subtree of a graph and the traveling salesman problem,”Proceedings of the American Mathematical Society 2 (1956) 48–50.Google Scholar
  11. [11]
    S. Lin, “Computer solution of the traveling salesman problem,”Bell System Technical Journal 44 (1965) 2245–2269.Google Scholar
  12. [12]
    T. Motzkin and I.J. Schoenberg, “The relaxation method for linear inequalities,”Canadian Journal of Mathematics 6 (1954) 393–404.Google Scholar

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

Personalised recommendations