Operations Research Letters 26 (2000) 27–32 www.elsevier.com/locate/orms An algorithm for the hierarchical Chinese postman problem Gianpaolo Ghiani a;b; ∗ , Gennaro Improta a a Dipartimento di Informatica e Sistemistica, Universit a “Federico II”, Via Claudio 21, 80125 Napoli, Italy b GERAD,  Ecole des Hautes  Etudes Commerciales, 3000, Chemin de la Cˆ ote-Sainte-Catherine, Montr eal, Qu ebec, Canada H3T 2A7 Received 1 July 1996; received in revised form 1 May 1999 Abstract The Hierarchical Chinese Postman Problem (HCPP) is a variant of the classical Chinese Postman Problem, in which the arcs are partitioned into clusters and a precedence relation is dened on clusters. Practical applications of the HCPP include snow and ice control on the roads and determination of optimal torch paths in ame cutting. The HCPP is NP-hard in general, but polynomial-time solvable if the precedence relation is linear and each cluster is connected. For this case an exact algorithm, requiring a lower computational eort than previous procedures, is described. c  2000 Elsevier Science B.V. All rights reserved. Keywords: Arc routing; Hierarchical Chinese postman problem 1. Introduction Let G(V; E) be an undirected graph, where V is the set of vertices, E is the set of edges and c ij (¿0) is the cost of traversing edge (v i ;v j ) ∈ E. The well-known Chinese Postman Problem (CPP) is to nd the shortest postman tour traversing each edge of E at least once. Several real-world situations, such as street sweeping, solid waste collection, mail delivery, meter reading, salt gritting and snow plowing can be modeled as a CPP subject to a set of side constraints [2,6,7]. In the Hierarchical Chinese Postman Problem (HCPP) a depot location d ∈ V is given, the edges are partitioned into clusters (or classes) and a prece- dence relation species the order in which the clusters are to be traversed starting from and ending at the depot. If the precedence relation is linear, the edge * Corresponding author. Fax: +39-081 7683636 set is partitioned into p classes E 1 ;E 2 ;:::;E p (E 1 ∪ E 2 ∪···∪ E p = E; E i ∩ E j = ∅∀i; j ∈{1;:::;p};i = j) and no edge in E i can be traversed before the traver- sal of all edges in E i−1 is completed. This variant of the HCPP has been used to model some practical problems: the determination of optimal torch paths in ame cutting [10] and snow removal operations in an urban or rural setting where streets are classied according to their importance (generally expressed by the average daily trac) [4,8]. Lemieux and Campagna [9] give simple heuristic procedures for a snow plowing problem in which streets are classied into main and secondary streets. Alfa and Liu [1] consider a weaker precedence rela- tion requiring that the traversal of E i starts before the beginning of the traversal of E i+1 and nishes before the ending of the traversal of E i+1 . Dror et al. [4] give a necessary and sucient condition for the exis- tence of a feasible solution and show that the HCPP 0167-6377/00/$ - see front matter c  2000 Elsevier Science B.V. All rights reserved. PII:S0167-6377(99)00046-2