Published Online:https://doi.org/10.1287/moor.2019.1002

We consider constrained versions of the prize-collecting traveling salesman and the prize-collecting minimum spanning tree problems. The goal is to maximize the number of vertices in the returned tour/tree subject to a bound on the tour/tree cost. Rooted variants of the problems have the additional constraint that a given vertex, the root, must be contained in the tour/tree. We present a 2-approximation algorithm for the rooted and unrooted versions of both the tree and tour variants. The algorithm is based on a parameterized primal–dual approach. It relies on first finding a threshold value for the dual variable corresponding to the budget constraint in the primal and then carefully constructing a tour/tree that is, in a precise sense, just within budget. We improve upon the best-known guarantee of 2 + ε for the rooted and unrooted tour versions and 3 + ε for the rooted and unrooted tree versions. Our analysis extends to the setting with weighted vertices, in which we want to maximize the total weight of vertices in the tour/tree. Interestingly enough, the algorithm and analysis for the rooted case and the unrooted case are almost identical.

INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.