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Transactions on Engineering Technologies pp 105–120Cite as

Arc Routing Based on the Zero-Suppressed Binary Decision Diagram

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Abstract

A wide range of problems in operations research falls under arc routing problems, a domain which focuses on arc or edge features rather than node or vertex attributes. The undirected rural postman problem is a well-known problem in arc routing that seeks to determine a minimum cost walk that traverses a certain set of required edges on a given graph. The problem, arising in numerous real-world applications, is nondeterministic polynomial-time hard. The chapter presents a solution to the undirected rural postman problem based on the zero-suppressed binary decision diagram, a compact data structure that represents a family of sets. Through an extension to the frontier-based search method of diagram construction, the approach solves the problem by efficient enumeration, producing all feasible routes in addition to the optimal route. Instances of the problem put forward in literature are then solved as benchmark for the decision-diagram-based solution. As reasonable time is consumed, the method proves to be a practicable candidate in solving the undirected rural postman problem.

Keywords

  • Arc routing
  • Enumeration algorithm
  • Frontier-based search
  • Graph optimization
  • Rural postman problem
  • Zero-suppressed binary decision diagram

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Notes

  1. 1.

    https://csacademy.com/app/graph_editor/.

  2. 2.

    https://github.com/kunisura/TdZdd.

  3. 3.

    https://github.com/renzopereztan/ZDDLines.

  4. 4.

    https://github.com/renzopereztan/URPDD.

References

  1. Angel, R., Caulde, W., Noonan, R., Whinston, A.: Computer-assisted school bus scheduling. Management Science pp. 279–288 (1972)

    Google Scholar 

  2. L. Bodin, S. Kursh, A computer-assisted system for the routing and scheduling of street sweepers. Operations Research 26(4), 528–537 (1978)

    CrossRef  Google Scholar 

  3. Braca, J., Bramel, J., Posner, B., Simchi-Levi, D.: A computerized approach to the new york city school bus routing project. Columbia University Working Paper (1993)

    Google Scholar 

  4. Christofides, N., Campos, V., Corberan, A., Mota, E.: An algorithm for the rural postman problem. Imperial College London Technical Report IC.OR.81.5 (1981)

    Google Scholar 

  5. Corberan, A., Sanchis, J.: A polyhedral approach for the rural postman problem. European Journal of Operations Research Vol. 79 Iss. 1 pp. 95–114 (1994)

    Google Scholar 

  6. de Cordoba, P.F., Garcia-Raffi, L., Sanchis, J.: A heuristic algorithm based on monte carlo methods for the rural postman problem. Computers and Operations Research Vol. 25 Iss. 12 pp. 1097–1106 (1998)

    Google Scholar 

  7. J. Desrosiers, J. Ferland, J. Rousseau, G. Lapalme, L. Chapleau, TRANSCOL: A multi-period school bus routing and scheduling system. Studies in the Management Sciences 22, 47–71 (1986)

    Google Scholar 

  8. J. Edmonds, E. Johnson, Matching, euler tours, and the chinese postman. Mathematical Programming 5(1), 88–124 (1973)

    MathSciNet  CrossRef  Google Scholar 

  9. H. Eiselt, M. Gendreau, G. Laporte, Arc routing problems, part ii: The rural postman problem. Operations Research 43(3), 399–414 (1995)

    CrossRef  Google Scholar 

  10. E. Fernandez, O. Meza, R. Garfinkel, M. Ortega, On the undirected rural postman problem: Tight bounds based on a new formulation. Operations Research 51(2), 281–291 (2003)

    MathSciNet  CrossRef  Google Scholar 

  11. G. Frederickson, M. Hecht, C. Kim, Approximation algorithms for some routing problems. SIAM Journal on Computing 7(2), 178–193 (1978)

    MathSciNet  CrossRef  Google Scholar 

  12. Ghiani, G., Laporte, G.: Arc Routing: Problems, Methods, and Applications, chap. The Undirected Rural Postman Problem, pp. 85–99. Society for Industrial and Applied Mathematics (2015)

    Google Scholar 

  13. E. Haslam, J. Wright, Applications of routing technologies to rural snow and ice control. Transportation Research Record No. 1304, 202–211 (1991)

    Google Scholar 

  14. Hertz, A.: Graph Theory, Combinatorics, and Algorithms, chap. Recent Trends in Arc Routing, pp. 215–236. Springer (2006)

    Google Scholar 

  15. A. Hertz, G. Laporte, P. Hugo, Improvement procedures for the undirected rural postman problem. INFORMS Journal on Computing 11(1), 53–62 (1999)

    MathSciNet  CrossRef  Google Scholar 

  16. Iwashita, H., Minato, S.: Efficient top-down ZDD construction techniques using recursive specifications. Hokkaido University Division of Computer Science TCS Technical Report TCS-TR-A-13-69 (2013)

    Google Scholar 

  17. Iwashita, H., Nakazawa, Y., Kawahara, J., Uno, T., Minato, S.: ZDD-based computation of the number of paths in a graph. Hokkaido University Division of Computer Science TCS Technical Report TCS-TR-A-12-60 (2012)

    Google Scholar 

  18. Kawahara, J., Inoue, T., Iwashita, H., Minato, S.: Frontier-based search for enumerating all constrained subgraphs with compressed representation. IEICE Transactions on Fundamentals of Electronics, Communications, and Computer Sciences Vol. E100-A No. 9 pp. 1773–1784 (2017)

    Google Scholar 

  19. Knuth, D.: The Art of Computer Programming Vol. 4 Fasc. 1. Addison-Wesley (2009)

    Google Scholar 

  20. Kreher, D., Stinson, D.: Combinatorial Algorithms: Generation, Enumeration, and Search. Chemical Rubber Company Press (2009)

    Google Scholar 

  21. J. Lenstra, A.R. Kan, On general routing problems. Networks 6(3), 273–280 (1976)

    MathSciNet  CrossRef  Google Scholar 

  22. S. Minato, Zero-suppressed BDDs and their applications. International Journal on Software Tools for Technology Transfer 3(2), 156–170 (2001)

    CrossRef  Google Scholar 

  23. Minato, S.: Zero-suppressed BDDs for set manipulation in combinatorial problems. Proceedings of the 30th International Design Automation Conference pp. 272–277 (Dallas, United States of America, 1993)

    Google Scholar 

  24. Moreira, M., Ferreira, J.: A genetic algorithm for the undirected rural postman problem (2010)

    Google Scholar 

  25. D. Morrison, E. Sewell, S. Jacobson, Solving the pricing problem in a branch-and-price algorithm for graph coloring using zero-suppressed binary decision diagrams. INFORMS Journal on Computing 28(1), 67–82 (2016)

    MathSciNet  CrossRef  Google Scholar 

  26. Orloff, C.: A fundamental problem in vehicle routing. Networks Vol. 4 Iss. 1 pp. 35–64 (1974)

    Google Scholar 

  27. Tan, R., Kawahara, J., Garciano, A., Sin, I.: A zero-suppressed binary decision diagram approach for constrained path enumeration. Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering 2019 pp. 132–136 (3-5 July 2019, London, United Kingdom)

    Google Scholar 

  28. T. Toda, T. Saitoh, H. Iwashita, J. Kawahara, S. Minato, ZDDs and enumeration problems: State-of-the-art techniques and programming tool. Computer Software 34(3), 97–120 (2017)

    Google Scholar 

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Acknowledgements

The research is supported by the Japan Society for the Promotion of Science through the Grants-in-Aid for Scientific Research Program (KAKENHI 18K19821). Valuable advice was received from Jun Kawahara. J. Kawahara is with the Kyoto University, Japan and the Nara Institute of Science and Technology, Japan. Further, Miguel Zenon Nicanor L. Saavedra has provided assistance in implementation. M. Z. N. L. Saavedra is with the Trainocate Group, Philippines and the Ateneo de Manila University, Philippines.

Author information

Authors and Affiliations

  1. Ateneo de Manila University, Quezon, Philippines

    Renzo Roel P. Tan

  2. Nara Institute of Science and Technology, Nara, Japan

    Renzo Roel P. Tan & Kazushi Ikeda

  3. Paris Sciences et Lettres University, Paris, France

    Florian Sikora

  4. Augmented Intelligence Pros, Incorporated, Manila, Philippines

    Kyle Stephen S. See

Authors
  1. Renzo Roel P. Tan
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  2. Florian Sikora
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  3. Kazushi Ikeda
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  4. Kyle Stephen S. See
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Corresponding author

Correspondence to Renzo Roel P. Tan .

Editor information

Editors and Affiliations

  1. International Association of Engineers, IAENG Sectretariat, Hong Kong, Hong Kong

    Dr. Sio-Iong Ao

  2. University of Huddersfield, Huddersfield, UK

    Prof. Len Gelman

  3. School of Software Convergence, Daegu Catholic University, Daegu, Korea (Republic of)

    Prof. Haeng Kon Kim

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Tan, R.R.P., Sikora, F., Ikeda, K., See, K.S.S. (2021). Arc Routing Based on the Zero-Suppressed Binary Decision Diagram. In: Ao, SI., Gelman, L., Kim, H.K. (eds) Transactions on Engineering Technologies. Springer, Singapore. https://doi.org/10.1007/978-981-15-8273-8_9

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  • DOI: https://doi.org/10.1007/978-981-15-8273-8_9

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