To fill or not to fill: the gas station problem

In this paper we study several routing problems that generalize shortest paths and the Traveling Salesman Problem. We consider a more general model that incorporates the actual cost in terms of gas prices. We have a vehicle with a given tank capacity. We assume that at each vertex gas may be purchased at a certain price. The objective is to find the cheapest route to go from s to t, or the cheapest tour visiting a given set of locations. Surprisingly, the problem of find the cheapest way to go from s to t can be solved in polynomial time and is not NP-complete. For most other versions however, the problem is NP-complete and we develop polynomial time approximation algorithms for these versions.