Optimal Cycles in Graphs and the Minimal Cost-To-Time Ratio Problem

  • Eugene L. Lawler
Part of the International Centre for Mechanical Sciences book series (CISM, volume 135)


Let G.= (N,A) be a directed graph, and let cij and t ij be a “cost” and a “transit time assigned to each arc (i, j). We consider three problems. The Negative Cycle Problem: Does G contain a directed cycle for which the sum of the costs is strictly negative? The Minimal-Cost Cycle Problem: As suming G contains no negative cycles, find a directed cycle for which the sum of the costs in minimum. The Minimal Cost-to-Time Ratio Cycle Problem: Find a directed cycle for which the sum of the costs divided by the sum of the transit times is minimum.

The first two problems can be solved by adaptations of well-know shortest path algorithms which require O (n3) computational steps, where n is the number of nodes of G. The minimal cost-to-time ratio problem can be solved by a procedure which uses a negative cycle algorithm as a subroutine. This procedure is essentially O (n3 log n) in complexity.

Various generalizations of these problems, in the form of side constraints on the cycles, are indicated.


Transit Time Binary Search Time Ratio Closed Path Directed Cycle 
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.


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  1. [1]
    R.E. Bellman, “On a routing Problem”, Quart. Appi. Math., XVI (1958) 87–90Google Scholar
  2. [2]
    A. Charnes and W.W. Cooper, “Programming with Linear Fractional Functions”, Naval Research Logistics Quarterly, 9 (1962) 181–186.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    G.B. Dantzig, W. Blattner and M.R. Rao, “Finfing a Cycle in a Graph with Minimum Cost to Time Ratio with Applications to a Ship Routing Problem”, Theory of Graphs, International Symposium, Dunod, Paris, and Gordon and Breach, New York (1966) 77–83.Google Scholar
  4. [4]
    S.E. Dreyfus, “An Appraisal of Some Shortest Path Algorithms”, Operations Research, 17 (May 1969) 395–412.CrossRefMATHGoogle Scholar
  5. [5]
    J. Edmonds, ‘Path, Trees, and Flowers“, Can. J. Math., 17 (1965) 449–467.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    M. Florian and P. Robert, “A Direct Search Method to Locate Negative Cycles in a Graph”, Management Science, 17 (1971) 307–310.Google Scholar
  7. [7]
    R.W. Floyd, ‘Algorithm 97, Shortest Path“, Comm. of the ACM, 5 (1962) 345.CrossRefGoogle Scholar
  8. [8]
    L.R. Ford, Jr, “Network Flow Theory”, the RAND corp., P-923 (Aug. 1965).Google Scholar
  9. [9]
    B.L. Fox, “Finding Minimal Cost-Time Ratio Circuits”, Operations Research, 17 (1969) 546–551.CrossRefGoogle Scholar
  10. [10]
    E.L. Lawler, “Optimal Cycles in Doubly Weighted Directed Linear Graphs’, Theory of Graphs, International Symposium, Dunod, Paris, and Gordon and Breach, New York (1966) 209–213.Google Scholar
  11. [11]
    E.F. Moore, “The Shortest Path Through a Maze”, Proc. Int. Symp. on the Theory of Switching, Part, II, April 1967, The Annals of the Computation Laboratory of Harvard University, 30, Harvard University Press (1959).Google Scholar
  12. [12]
    F. Shapiro, “Shortest Route Methods for Finite State Space Dynamic Programming Problems”, SIAM J. Applied Math, 16 (1968) 1232–1250.MATHGoogle Scholar
  13. [13]
    I:.L. Traiger and A. Gill, “On an Asymptotic Optimization Problem in Finite, Directed, Weighted Graphs,” Information and Control, 13 (1968) 527–533.Google Scholar
  14. [14]
    J.Y. Yen, “An Algorithm for Finding Shortest Routes from All Source Nodes to Given Destination in General Networks”, Quart. of Appl. Math., 27, (Jan. 1970) 526–530.MATHGoogle Scholar
  15. [15]
    J. Y. Yen, “A Shortest Path Algorithm’, Ph.D. Dissertation, University of California, Berkeley, California (1970).Google Scholar

Copyright information

© Springer-Verlag Wien 1972

Authors and Affiliations

  • Eugene L. Lawler
    • 1
  1. 1.Department of Electrical EngineeringUniversity of Colorado formally with the Polytechnic Institute of BrooklynUSA

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