On crossings, the Crossing Postman Problem, and the Rural Postman Problem

Garfinkel, Robert; Webb, Ian R.

doi:10.1002/(sici)1097-0037(199910)34:3<173::aid-net1>3.0.co;2-w

Cited by 24 publications

(15 citation statements)

References 9 publications

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“…It is known that the RPP reduces to the CPP when the graph G R is connected, and this is also true for the XPP . It is easy to see that, in this case, the Drone RPP also reduces to the CPP in the sense that its optimal solution is found by obtaining a minimum cost matching on the odd‐degree vertices.…”

Section: Drones and Arps With A Single Vehiclementioning

confidence: 90%

“…Note that Definition is not restricted to a plane, though for clarity our illustrations will be planar. A simpler version of this problem was proposed in Garfinkel and Webb , who called it the crossing postman problem (XPP). As in the Drone RPP, in the XPP it is permitted to leave the edges of the network and travel from one edge to another at points other than the original vertices.…”

Section: Drones and Arps With A Single Vehiclementioning

confidence: 99%

“…A standard RPP instance with these characteristics may be very hard to solve optimally. However, due to its special structure, the Drone RPP instance may not be as difficult to solve as it might seem for the following reasons: (a)Recall that the difficulty of solving the RPP depends strongly on the number of R ‐connected components (the connected components of graph G R = ( V R , E R )), and that the Drone RPP instance has the same number of R ‐connected components as the initial problem. (b) The (potentially huge) number of nonrequired edges can be reduced as the following nonrequired edges can be removed: Nonrequired edges joining two vertices in the same R ‐connected component, except the edges joining two odd‐degree vertices (see Lemma 3 by Garfinkel and Webb in ). The reason is as follows: Consider a feasible solution deadheading an edge ( i , j ), with vertices i , j in the same R ‐connected component and j being an even‐degree vertex.…”

Section: Drones and Arps With A Single Vehiclementioning

confidence: 99%

“…In addition to having several vehicles, the length constraints are difficult to handle. Furthermore, the number of nonrequired edges cannot be reduced as much as in the Drone RPP, because some of the rules described by Garfinkel and Webb for the XPP and the RPP cannot be applied here. For example, the nonrequired edges joining two vertices, not both with odd degree, of the same R ‐connected component cannot be removed.…”

Section: Drones and Arps With Several Vehiclesmentioning

confidence: 99%

“…Dewil et al classifies cutting path problems into six types, and the one which most resembles the problem studied in this paper is the Intermittent Cutting Problem (ICP), in which the object can be cut in sections and there is no restriction on the points that can be used for entry or exit. The only two works reported in related to the ICP are those by Garfinkel and Webb , which will be discussed later, and Rodrigues and Soeiro , which proposes a memetic algorithm for the RPP and applies it to the solution of a cutting path problem appearing in an industrial application.…”

Section: Introductionmentioning

confidence: 99%

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Drone arc routing problems

et al. 2018

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In this article, we present some drone arc routing problems (Drone ARPs) and study their relation with well‐known postman ARPs. Applications for Drone ARPs include traffic monitoring by flying over roadways, infrastructure inspection such as by flying along power transmission lines, pipelines or fences, and surveillance along linear features such as coastlines or territorial borders. Unlike the postmen in traditional ARPs, drones can travel directly between any two points in the plane without following the edges of the network. As a consequence, a drone route may service only part of an edge, with multiple routes being used to cover the entire edge. Thus the Drone ARPs are continuous optimization problems with an infinite number of feasible solutions. In order to solve them as a discrete optimization problem, we approximate each curve in the plane by a polygonal chain, thus allowing the vehicle to enter and leave each curve only at the points of the polygonal chain. If the capacity of the vehicles is unlimited, the resulting problem is a rural postman problem (RPP). We propose an algorithm that iteratively solves RPP instances with an increasing number of points of the polygonal chain and present results on several sets of instances. We also briefly discuss the case in which the drones have limited capacity and several drones are needed.

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Section: Drones and Arps With A Single Vehiclementioning

confidence: 90%

Section: Drones and Arps With A Single Vehiclementioning

confidence: 99%

Section: Drones and Arps With A Single Vehiclementioning

confidence: 99%

Section: Drones and Arps With Several Vehiclesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Drone arc routing problems

et al. 2018

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Recent results on Arc Routing Problems: An annotated bibliography

Corberán

Prins

2010

Networks

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Arc routing problems are vehicle routing problems in which the tasks to be performed are located on some arcs or edges of a network, e.g., spreading salt on a road or collecting municipal refuse in a street. They have been much less studied than node routing problems but impressive developments have been achieved in the last decade. The objective of this article is to survey these recent results and to provide an annotated bibliography. Except for a few references, the period surveyed starts from the publication of the book "Arc routing: theory, solutions and applications", edited by Dror in 2000. The article is made up of two main parts: one on uncapacitated (single vehicle) problems derived from the Chinese Postman Problem, the Rural Postman Problem, and the General Routing Problem; and one on multivehicle or capacitated problems such as the Capacitated Arc Routing Problem.

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Arc routing problems: A review of the past, present, and future

et al. 2020

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Arc routing problems (ARPs) are defined and introduced. Following a brief history of developments in this area of research, different types of ARPs are described that are currently relevant for study. In addition, particular features of ARPs that are important from a theoretical or practical point of view are discussed. A section on applications describes some of the changes that have occurred from early applications of ARP models to the present day and points the way to emerging topics for study. A final section provides information on libraries and instance repositories for ARPs. The review concludes with some perspectives on future research developments and opportunities for emerging applications.

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On crossings, the Crossing Postman Problem, and the Rural Postman Problem

Cited by 24 publications

References 9 publications

Drone arc routing problems

Drone arc routing problems

Recent results on Arc Routing Problems: An annotated bibliography

Arc routing problems: A review of the past, present, and future

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