The Generalized Covering Salesman Problem

Abstract

Abstract:Given a graph ( , ) G N E  , the Covering Salesman Problem (CSP) is to identify the minimum length tour "covering" all the nodes. More specifically, it seeks the minimum length tour visiting a subset of the nodes in N such that each node i not on the tour is within a predetermined distance d i of a node on the tour. In this paper, we define and develop a generalized version of the CSP, and refer to it as the Generalized Covering Salesman Problem (GCSP). Here, each node i needs to be covered at least i k times and there is a cost associated with visiting each node. We seek a minimum cost tour such that each node i is covered at least i k times by the tour. We define three variants of the GCSP. In the first case, each node can be visited by the tour at most once. In the second version, visiting a node i more than once is possible, but an overnight stay is not allowed (i.e., to revisit a node i, the tour has to visit another node before it can return to i). Finally, in the third variant, the tour can visit each node more than once consecutively. In this paper, we develop two local search heuristics to find high-quality solutions to the three GCSP variants. In order to test the proposed algorithms, we generated datasets based on TSP Library instances. Since the CSP and the Generalized Traveling Salesman Problem are special cases of the GCSP, we tested our heuristics on both of these two problems as well. Overall, the results show that our proposed heuristics find highquality solutions very rapidly.

Optimization Approaches for the Traveling Salesman Problem with Drone

“…Several generalizations and special cases of the CSP have been studied in the literature, see e.g. Gulczynski et al [2006], Shuttleworth et al [2008], Golden et al [2012], Behdani and Smith [2014]. Similar to the CSP, in the TSP-D the truck does not have to visit all nodes.…”

Combining ant colony optimization algorithm and dynamic programming technique for solving the covering salesman problem

“…In case of having multiple optimal solutions for the SCP, all of them are used in the second step and the least costly TSP tour is selected. Golden, Naji-Azimi, Raghavan, Salari, and Toth (2012) proposed two local search algorithms for the CSP. Their proposed algorithms take advantage of several moves like swap, removal-reinsertion, and perturbation to escape local optimal solutions.…”

“…In this problem, the classification of the vertices is the same as that defined for the CTP, while the only difference is that we have to cover the demand using more than one vehicle. Golden et al (2012) proposed some natural generalizations of the CSP, where the assumption of visiting or covering a vertex for only one time is relaxed. They referred to some applications arising in rural healthcare delivery, for which we have to visit or cover a vertex for more than once to satisfy its corresponding demand.…”

Efficient measurement planning for remote gas sensing with mobile robots

“…Ignoring the cost associated to sensing actions, our problem could be reduced to the Generalized Covering Salesman Problem [8], whose instances are solved by generating paths from which the whole environment can be observed. A similar problem is solved in [9], where an approach based on mixed integer linear programming is used for finding a surveillance route for a mobile camera.…”