The Generalized Covering Salesman Problem

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 we refer to it as the generalized covering salesman problem (GCSP). Here, each node i needs to be covered at least k(i) 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 k(i) 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 case, 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 case, 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. To test the proposed algorithms, we generated data sets based on traveling salesman problem library instances. Because the CSP and the generalized traveling salesman problem are special cases of the GCSP, we tested our heuristics on both of those problems as well. Overall, the results show that our proposed heuristics find high-quality solutions very rapidly.