New results on the Windy Postman Problem

Abstract

In this paper we study the Windy Postman Problem. This is a well known Arc Routing Problem which contains the Mixed Chinese Postman Problem as a special case. We present here a full description of the polyhedron associated to the Windy Postman Problem over graphs with up to 4 vertices and 10 edges. We introduce two new families of facet-inducing inequalities and we prove that these inequalities and the already known Odd Zigzag inequalities are mod-2 cut inequalities. Moreover, a Branch & Cut (B&C) 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 the B&C algorithm over several sets of large WPP and MCPP instances, with up to 3000 nodes and 9000 links, shows that up to our knowledge this is the best algorithm for the exact resolution of the WPP and the MCPP.

Solving the equality generalized traveling salesman problem using the Lin–Kernighan–Helsgaun Algorithm

“…However, this will probably be at the expense of unacceptable running times. For these large instances, GLKH cannot compete with the highly sophisticated exact algorithm of Corberán et al [20]. • Tests similar to those reported in Table 6 have been conducted by Drexl [21, p. 10].…”

An updated annotated bibliography on arc routing problems

“…Although usually NP‐hard, the Windy CPP has some polynomial special cases, as over Eulerian networks (e.g., Refs. and ).…”

“…A 3-regular graph is a graph where all nodes share the same degree, it being equal to 3. Moreover, solving [45,124] Including TW constraints [124] Generalized or close-enough [124] CPP solutions on 3-regular (multi)graphs [186] With load-dependent costs -the cost of traversing an edge depends on its length and on the weight of the vehicle's cargo [61] Edge-colored multigraph -each edge has a color, and a tour is called properly colored if no two consecutive edges share the same color [45,68,194] MCPP Min(cost) 100,194] Lower and upper bounds on the number of times each link is traversed [76] WCPP Min(cost) W A R = ∅ E R = E N R = ∅ Costs on links depend on the travel direction [66,68,194] HCPP Min(cost) U A R = ∅ E R = E N R = ∅ Links in a higher hierarchic level need to be serviced before links in lower levels [124] TDCPP Min(cost) D A R = A E R = ∅ N R = ∅ Time-dependent -the time needed to service a link depends on the service start time [187,190] (Continued)…”

Recent results on Arc Routing Problems: An annotated bibliography

“…In this article, it is also proved that these inequalities generalize the 3-wheel inequalities proposed by Win [143] and that they play an important role in the exact resolution of the WPP. In Corberán et al [48], a branch-and-cut procedure able to solve large size instances of up to 3000 nodes and 9000 edges is described. It is based on a cutting-plane algorithm that identifies violated odd-cut and odd zigzag inequalities and two other new facetinducing inequality types.…”

“…The branch-and-cut algorithm proposed in Corberán et al [48], which is briefly described in the next section, is able to solve large size instances. Specifically, 17 of 24 instances with 3000 nodes and up to 9000 arcs and edges were solved to optimality without using a heuristic producing initial upper bounds.…”