Abstract
We provide a new characterization of the NP-hard arc routing problem Rural Postman in terms of a constrained variant of minimum-weight perfect matching on bipartite graphs. To this end, we employ a parameterized equivalence between Rural Postman and Eulerian Extension, a natural arc addition problem in directed multigraphs. We indicate the NP-hardness of the introduced matching problem. In particular, we use it to make some partial progress towards answering the open question about the parameterized complexity of Rural Postman with respect to the number of weakly connected components in the graph induced by the required arcs. This is a more than thirty years open and long-time neglected question with significant practical relevance.
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References
Assad, A.A., Golden, B.L.: Arc routing methods and applications. In: Network Routing. Handbooks in Operations Research and Management Science, vol. 8, pp. 375–483. Elsevier B.V. (1995)
Benavent, E., Corberán, A., Piñana, E., Plana, I., Sanchis, J.M.: New heuristic algorithms for the windy rural postman problem. Comput. Oper. Res. 32(12), 3111–3128 (2005)
Cabral, E.A., Gendreau, M., Ghiani, G., Laporte, G.: Solving the hierarchical chinese postman problem as a rural postman problem. European J. Oper. Res. 155(1), 44–50 (2004)
Dorn, F., Moser, H., Niedermeier, R., Weller, M.: Efficient Algorithms for Eulerian Extension. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 100–111. Springer, Heidelberg (2010)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Dror, M.: Arc Routing: Theory, Solutions, and Applications. Kluwer Academic Publishers (2000)
Eiselt, H.A., Gendreau, M., Laporte, G.: Arc routing problems, part II: The rural postman problem. Oper. Res. 43(3), 399–414 (1995)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Frederickson, G.N.: Approximation algorithms for some postman problems. J. ACM 26(3), 538–554 (1979)
Höhn, W., Jacobs, T., Megow, N.: On Eulerian extensions and their application to no-wait flowshop scheduling. J. Sched (to appear, 2011)
Lenstra, J.K., Kan, A.H.G.R.: On general routing problems. Networks 6(3), 273–280 (1976)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)
Orloff, C.S.: A fundamental problem in vehicle routing. Networks 4(1), 35–64 (1974)
Orloff, C.S.: On general routing problems: Comments. Networks 6(3), 281–284 (1976)
Perrier, N., Langevin, A., Campbell, J.F.: A survey of models and algorithms for winter road maintenance. Part IV: Vehicle routing and fleet sizing for plowing and snow disposal. Comput. Oper. Res. 34(1), 258–294 (2007)
Sorge, M.: On making directed graphs Eulerian. Diplomarbeit, Institut für Informatik, Friedrich-Schiller-Universität Jena (2011), Available electronically. arXiv:1101.4283 [cs.DM]
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Sorge, M., van Bevern, R., Niedermeier, R., Weller, M. (2011). A New View on Rural Postman Based on Eulerian Extension and Matching. In: Iliopoulos, C.S., Smyth, W.F. (eds) Combinatorial Algorithms. IWOCA 2011. Lecture Notes in Computer Science, vol 7056. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25011-8_25
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DOI: https://doi.org/10.1007/978-3-642-25011-8_25
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