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.
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References
Angel, R., Caulde, W., Noonan, R., Whinston, A.: Computer-assisted school bus scheduling. Management Science pp. 279–288 (1972)
L. Bodin, S. Kursh, A computer-assisted system for the routing and scheduling of street sweepers. Operations Research 26(4), 528–537 (1978)
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)
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)
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)
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)
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)
J. Edmonds, E. Johnson, Matching, euler tours, and the chinese postman. Mathematical Programming 5(1), 88–124 (1973)
H. Eiselt, M. Gendreau, G. Laporte, Arc routing problems, part ii: The rural postman problem. Operations Research 43(3), 399–414 (1995)
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)
G. Frederickson, M. Hecht, C. Kim, Approximation algorithms for some routing problems. SIAM Journal on Computing 7(2), 178–193 (1978)
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)
E. Haslam, J. Wright, Applications of routing technologies to rural snow and ice control. Transportation Research Record No. 1304, 202–211 (1991)
Hertz, A.: Graph Theory, Combinatorics, and Algorithms, chap. Recent Trends in Arc Routing, pp. 215–236. Springer (2006)
A. Hertz, G. Laporte, P. Hugo, Improvement procedures for the undirected rural postman problem. INFORMS Journal on Computing 11(1), 53–62 (1999)
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)
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)
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)
Knuth, D.: The Art of Computer Programming Vol. 4 Fasc. 1. Addison-Wesley (2009)
Kreher, D., Stinson, D.: Combinatorial Algorithms: Generation, Enumeration, and Search. Chemical Rubber Company Press (2009)
J. Lenstra, A.R. Kan, On general routing problems. Networks 6(3), 273–280 (1976)
S. Minato, Zero-suppressed BDDs and their applications. International Journal on Software Tools for Technology Transfer 3(2), 156–170 (2001)
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)
Moreira, M., Ferreira, J.: A genetic algorithm for the undirected rural postman problem (2010)
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)
Orloff, C.: A fundamental problem in vehicle routing. Networks Vol. 4 Iss. 1 pp. 35–64 (1974)
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)
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)
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.
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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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