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The optimum traversal of a graph

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  • Christofides, Nicos

Abstract

For a given graph (network) having costs [cij] associated with its links, the present paper examines the problem of finding a cycle which traverses every link of the graph at least once, and which incurs the minimum cost of traversal. This problem (called thegraph traversal problem, or theChinese postman problem [9]) can be formulated in ways analogous to those used for the well-known travelling salesman problem, and using this apparent similarity, Bellman and Cooke [1] have produced a dynamic programming formulation. This method of solution of the graph traversal problem requires computational times which increase exponentially with the number of links in the graph. Approximately the same rate of increase of computational effort with problem size would result by any other method adapting a travelling salesman algorithm to the present problem. This paper describes an efficient algorithm for the optimal solution of the graph traversal problem based on the matching method of Edmonds [5, 6]. The computational time requirements of this algorithm increase as a low order (2 or 3) power of the number of links in the graph. Computational results are given for graphs of up to 50 vertices and 125 links. The paper then discusses a generalised version of the graph traversal problem, where not one but a number of cycles are required to traverse the graph. In this case each link has (in addition to its cost) a quantity qij associated with it, and the sum of the quantities of the links in any one cycle must be less than a given amount representing the cycle capacity. A heuristic algorithm for the solution of this problem is given. The algorithm is based on the optimal algorithm for the single-cycle graph traversal problem and is shown to produce near-optimal results. There is a large number of possible applications where graph traversal problems arise. These applications include: the spraying of roads with salt-grit to prevent ice formation, the inspection of electric power lines, gas, or oil pipelines for faults, the delivery of letter post, etc.

Suggested Citation

  • Christofides, Nicos, 1973. "The optimum traversal of a graph," Omega, Elsevier, vol. 1(6), pages 719-732, December.
  • Handle: RePEc:eee:jomega:v:1:y:1973:i:6:p:719-732
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    Cited by:

    1. Corberan, A. & Marti, R. & Sanchis, J. M., 2002. "A GRASP heuristic for the mixed Chinese postman problem," European Journal of Operational Research, Elsevier, vol. 142(1), pages 70-80, October.
    2. Sullivan, James L. & Dowds, Jonathan & Novak, David C. & Scott, Darren M. & Ragsdale, Cliff, 2019. "Development and application of an iterative heuristic for roadway snow and ice control," Transportation Research Part A: Policy and Practice, Elsevier, vol. 127(C), pages 18-31.
    3. Amberg, Anita & Domschke, Wolfgang & Vo[ss], Stefan, 2000. "Multiple center capacitated arc routing problems: A tabu search algorithm using capacitated trees," European Journal of Operational Research, Elsevier, vol. 124(2), pages 360-376, July.
    4. Santos, Lui­s & Coutinho-Rodrigues, João & Current, John R., 2008. "Implementing a multi-vehicle multi-route spatial decision support system for efficient trash collection in Portugal," Transportation Research Part A: Policy and Practice, Elsevier, vol. 42(6), pages 922-934, July.
    5. Alain Hertz & Michel Mittaz, 2001. "A Variable Neighborhood Descent Algorithm for the Undirected Capacitated Arc Routing Problem," Transportation Science, INFORMS, vol. 35(4), pages 425-434, November.
    6. Andie Pramudita & Eiichi Taniguchi, 2014. "Model of debris collection operation after disasters and its application in urban area," International Journal of Urban Sciences, Taylor & Francis Journals, vol. 18(2), pages 218-243, July.
    7. Wei Yu & Zhaohui Liu & Xiaoguang Bao, 2021. "Approximation algorithms for some min–max postmen cover problems," Annals of Operations Research, Springer, vol. 300(1), pages 267-287, May.
    8. Tagmouti, Mariam & Gendreau, Michel & Potvin, Jean-Yves, 2007. "Arc routing problems with time-dependent service costs," European Journal of Operational Research, Elsevier, vol. 181(1), pages 30-39, August.
    9. Xiaoguang Bao & Xinhao Ni, 2024. "Approximation algorithms for two clustered arc routing problems," Journal of Combinatorial Optimization, Springer, vol. 47(5), pages 1-12, July.
    10. Alain Hertz & Gilbert Laporte & Michel Mittaz, 2000. "A Tabu Search Heuristic for the Capacitated arc Routing Problem," Operations Research, INFORMS, vol. 48(1), pages 129-135, February.

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