Temporal Orienteering with Changing Fuel Costs

TL;DR

This paper introduces a new variant of the orienteering problem where travel costs depend on departure and arrival times, modeling real-world scenarios like space travel with time-dependent fuel costs.

Contribution

It generalizes temporal graphs to include time-dependent edge costs and explores algorithmic solutions under various restrictions.

Findings

Deciding feasibility is NP-complete.

Efficient algorithms exist under specific restrictions.

The model is likely of independent interest.

Abstract

The problem Orienteering asks whether there exists a walk which visits a number of sites without exceeding some fuel budget. In the variant of the problem we consider, the cost of each edge in the walk is dependent on the time we depart one endpoint and the time we arrive at the other endpoint. This mirrors applications such as travel between orbiting objects where fuel costs are dependent on both the departure time and the length of time spent travelling. In defining this problem, we introduce a natural generalisation of the standard notion of temporal graphs: the pair consisting of the graph of the sites and a cost function, in which costs as well as shortest travel times between pairs of objects change over time. We believe this model is likely to be of independent interest. The problem of deciding whether a stated goal is feasible is easily seen to be NP-complete; we investigate three different ways to restrict the input which lead to efficient algorithms. These include the number of times an edge can be used, an analogue of vertex-interval-membership width, and the number of sites to be visited.