New results on the Windy Postman Problem
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In this paper, we study the Windy Postman Problem (WPP). This is a well-known Arc Routing Problem that contains the Mixed Chinese Postman Problem (MCPP) as a special case. We extend to arbitrary dimension some new inequalities that complete the description of the polyhedron associated with the Windy Postman Problem over graphs with up to four vertices and ten edges. We introduce two new families of facet-inducing inequalities and prove that these inequalities, along with the already known odd zigzag inequalities, are Chvátal–Gomory inequalities of rank at most 2. Moreover, a branch-and-cut algorithm that incorporates two new separation algorithms for all the previously mentioned inequalities and a new heuristic procedure to obtain upper bounds are presented. Finally, the performance of a branch-and-cut algorithm over several sets of large WPP and MCPP instances, with up to 3,000 nodes and 9,000 edges (and arcs in the MCPP case), shows that, to our knowledge, this is the best algorithm to date for the exact resolution of the WPP and the MCPP.
KeywordsPolyhedral combinatorics Facets Arc routing Windy Postman Problem Mixed Chinese Postman Problem
Mathematics Subject Classification (2000)90C57
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- 2.Brucker P. The Chinese postman problem for mixed graphs. In Proceedings of international workshop. Lecture Notes in Computer Science 100, 354–366 (1981)Google Scholar
- 5.Christof, T., Loebel, A.: PORTA—a polyhedron representation algorithm www.informatik.uni-heidelberg.de/groups/comopt/software/PORTA/ (1998)
- 6.Christofides, N., Benavent, E., Campos, V., Corberán, A., Mota, E.: An optimal method for the mixed postman problem. In Thoft-Christensen, P. (ed.) System Modelling and Optimization. Lecture Notes in Control and Information Sciences 59, Springer (1984)Google Scholar
- 9.Corberán, A., Plana I., Sanchis, J.M.: Arc routing problems: data instances. www.uv.es/corberan/instancias.htm (2007)
- 11.Grötschel, M., Win, Z.: On the windy postman polyhedron. Report No. 75, Schwerpunktprogram der Deutschen Forschungsgemeinschaft, Universität Augsburg, Germany (1988)Google Scholar
- 19.Wenger, K.: Generic Cut Generation Methods for Routing Problems. PhD Dissertation, University of Heidelberg, Germany (2004)Google Scholar
- 20.Win, Z.: Contributions to Routing Problems. PhD Dissertation, University of Augsburg, Germany (1987)Google Scholar