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New results on the Windy Postman Problem

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Abstract

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.

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Authors and Affiliations

  1. Departament d’Estadística i Investigació Operativa, Universitat de València, Valencia, Spain

    Angel Corberán

  2. Institute of Computer Science, University of Heidelberg, Heidelberg, Germany

    Marcus Oswald & Gerhard Reinelt

  3. Departament de Matemàtiques per a l’Economia i l’Empresa, Universitat de València, Valencia, Spain

    Isaac Plana

  4. Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, Valencia, Spain

    José M. Sanchis

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  1. Angel Corberán
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  2. Marcus Oswald
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  3. Isaac Plana
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  4. Gerhard Reinelt
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  5. José M. Sanchis
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Correspondence to Angel Corberán.

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Corberán, A., Oswald, M., Plana, I. et al. New results on the Windy Postman Problem. Math. Program. 132, 309–332 (2012). https://doi.org/10.1007/s10107-010-0399-x

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  • DOI: https://doi.org/10.1007/s10107-010-0399-x

Keywords

Mathematics Subject Classification (2000)