Orienteering (with Time Windows) on Restricted Graph Classes

Given a graph with edge costs and vertex profits and given a budget B, the Orienteering Problem asks for a walk of cost at most B of maximum profit. Additionally, each profit may be given with a time window within it can be collected by the walk. While the Orienteering Problem and thus the version with time windows are NP-hard in general, it remains open on numerous special graph classes. Since in several applications, especially for planning a route from A to B with waypoints, the input graph can be restricted to tree-like or path-like structures, in this paper we consider orienteering on these graph classes. While the Orienteering Problem with time windows is NP-hard even on undirected paths and cycles, and remains so even if all profits must be collected, we show that for directed paths it can be solved efficiently, even if each profit can be collected in one of several time windows. The same case is shown to be NP-hard for directed cycles. Particularly interesting is the Orienteering Problem on a directed cycle with one time window per profit. We give an efficient algorithm for the case where all time windows are shorter than the length of the cycle, resulting in a 2-approximation for the general setting. We further develop a polynomial-time approximation scheme for this problem and give a polynomial algorithm for the case where all profits must be collected. For the Orienteering Problem with time windows for the edges, we give a quadratic time algorithm for undirected paths and observe that the problem is NP-hard for trees. In the variant without time windows, we show that on trees and thus on graphs with bounded tree-width the Orienteering Problem remains NP-hard. We present, however, an FPT algorithm to solve orienteering with unit profits that we then use to obtain a ($1+\varepsilon$)-approximation algorithm on graphs with arbitrary profits and bounded tree-width.