Abstract
In the Mixed Chinese Postman Problem (MCPP), given an edge-weighted mixed graph G (G may have both edges and arcs), our aim is to find a minimum weight closed walk traversing each edge and arc at least once. The MCPP parameterized by the number of edges was known to be fixed-parameter tractable using a simple argument. Solving an open question of van Bevern et al., we prove that the MCPP parameterized by the number of arcs is also fixed-parameter tractable. Our proof is more involved and, in particular, uses a very useful result of Marx, O’Sullivan and Razgon (2013) on the treewidth of torso graphs with respect to small separators.
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References
Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications, 2nd edn. Springer (2009)
van Bevern, R., Niedermeier, R., Sorge, M., Weller, M.: Complexity of Arc Rooting Problems. In: Corberán, A., Laporte, G. (eds.) Arc Routing: Problems, Methods and Applications, ch. 2. SIAM, Phil
Beltrami, E.J., Bodin, L.D.: Networks and vehicle routing for municipal waste collection. Networks 4(1), 65–94 (1974)
Brucker, P.: The Chinese postman problem for mixed graphs. In: Noltemeier, H. (ed.) Graphtheoretic Concepts in Computer Science. LNCS, vol. 100, pp. 354–366. Springer, Heidelberg (1981)
Christofides, N.: The optimum traversal of a graph. Omega 1, 719–732 (1973)
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer (2013)
Edmonds, J., Johnson, E.L.: Matching, Euler tours and the Chinese postman. Mathematical Programming 5, 88–124 (1973)
Eiselt, H.A., Gendreau, M., Laporte, G.: Arc routing problems. I. The Chinese postman problem. Oper. Res. 43, 231–242 (1995)
Fernandes, C.G., Lee, O., Wakabayashi, Y.: Minimum cycle cover and Chinese postman problems on mixed graphs with bounded tree-width. Discrete Applied Mathematics 157(2), 272–279 (2009)
Gutin, G., Jones, M., Sheng, B.: Parameterized Complexity of the k-Arc Chinese Postman Problem, http://arxiv.org/abs/1403.1512
Marx, D., O’Sullivan, B., Razgon, I.: Finding small separators in linear time via treewidth reduction. ACM Trans. Algorithms 9, article 30 (2013)
Minieka, E.: The Chinese postman problem for mixed networks. Management Sci. 25, 643–648 (1979/80)
Papadimitriou, C.H.: On the complexity of edge traversing. J. ACM 23, 544–554 (1976)
Peng, Y.: Approximation algorithms for some postman problems over mixed graphs. Chinese J. Oper. Res. 8, 76–80 (1989)
Sorge, M.: Some Algorithmic Challenges in Arc Routing. In: Talk at NII Shonan Seminar, no. 18 (May 2013)
Zaragoza Martínez, F.J.: Postman Problems on Mixed Graphs. PhD thesis, University of Waterloo (2003)
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Gutin, G., Jones, M., Sheng, B. (2014). Parameterized Complexity of the k-Arc Chinese Postman Problem. In: Schulz, A.S., Wagner, D. (eds) Algorithms - ESA 2014. ESA 2014. Lecture Notes in Computer Science, vol 8737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44777-2_44
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