Discrete Optimization Branch-and-cut algorithms for the undirected m-Peripatetic Salesman Problem  Eric Duchenne a , Gilbert Laporte b, * ,Fr ed eric Semet a a LAMIH-ROI, Universit e de Valenciennes, Le Mont Houy, 59313 Valenciennes Cedex 9, France b GERAD and Canada Research Chair in Distribution Management, HEC Montr eal, 3000 chemin de la C^ ote-Sainte-Catherine, Montreal, Que., Canada H3T 2A7 Received 3 February 2003; accepted 23 September 2003 Available online 13 December 2003 Abstract In the m-Peripatetic Salesman Problem (m-PSP) the aim is to determine m edge disjoint Hamiltonian cycles of minimum total cost on a graph. This article describes exact branch-and-cut solution procedures for the undirected m- PSP. Computational results are reported on random and Euclidean graphs. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Peripatetic Salesman Problem; Traveling Salesman Problem; Branch-and-cut 1. Introduction The m-Peripatetic Salesman Problem (m-PSP) is defined on a complete graph G ¼ðV ; EÞ, where V ¼f1; ... ; ng is a vertex set and E ¼ fði; jÞ : i; j 2 V ; i < jg is an edge set. A cost matrix C ¼ðc ij Þ is defined on E. The problem consists of determining m edge disjoint Hamiltonian cycles of minimum total cost on G. When m ¼ 1 the m-PSP reduces to the Traveling Salesman Problem (TSP). In the sequel we assume that m < ðn  1Þ=2 to avoid trivial or infeasible cases. The m-PSP was introduced by Krarup (1975). Applications include the design of watchman tours (WolfterCalvoandCordone,2003)whereitisoftenimportanttoassignasetofedge-disjointroundstothe watchman in order to avoid always repeating the same tour and thus enhance security. In the same spirit, DeKort(1993)citesanetworkdesignapplicationwhere,inordertoprotectthenetworkfromlinkfailure, severaledges-disjointcyclesmustbedetermined.Thisauthoralsomentionsaschedulingapplicationofthe 2-PSP where each job must be processed twice by the same machine but technological constraints prevent the repetition of identical job sequences. * Corresponding author. Address: Centre de Recherche sur les Transports, Universite de Montreal, Ecole des Hautes Etudes Commerciales, Succursale Centre-Ville, C.P. 6128, Montreal, Que., Canada H3C 3J7. Tel.: +1-514-343-6143; fax: +1-514-343-7121. E-mail address: gilbert@crt.umontreal.ca (G. Laporte). 0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2003.09.024 European Journal of Operational Research 162 (2005) 700–712 www.elsevier.com/locate/dsw