Mathematical Programming

, Volume 132, Issue 1–2, pp 309–332 | Cite as

New results on the Windy Postman Problem

  • Angel Corberán
  • Marcus Oswald
  • Isaac Plana
  • Gerhard Reinelt
  • José M. Sanchis
Full Length Paper Series A

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.

Keywords

Polyhedral combinatorics Facets Arc routing Windy Postman Problem Mixed Chinese Postman Problem 

Mathematics Subject Classification (2000)

90C57 

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Copyright information

© Springer and Mathematical Optimization Society 2010

Authors and Affiliations

  • Angel Corberán
    • 1
  • Marcus Oswald
    • 2
  • Isaac Plana
    • 3
  • Gerhard Reinelt
    • 2
  • José M. Sanchis
    • 4
  1. 1.Departament d’Estadística i Investigació OperativaUniversitat de ValènciaValenciaSpain
  2. 2.Institute of Computer ScienceUniversity of HeidelbergHeidelbergGermany
  3. 3.Departament de Matemàtiques per a l’Economia i l’EmpresaUniversitat de ValènciaValenciaSpain
  4. 4.Departamento de Matemática AplicadaUniversidad Politécnica de ValenciaValenciaSpain

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