A fast algorithm for the gas station problem

In the gas station problem we want to find the cheapest path between two vertices of an $n$-vertex graph. Our car has a specific fuel capacity and at each vertex we can fill our car with gas, with the fuel cost depending on the vertex. Furthermore, we are allowed at most $\Delta$ stops for refuelling. In this short paper we provide an algorithm solving the problem in $O(\Delta n^2 + n^2\log{n})$ steps improving an earlier result by Khuller, Malekian and Mestre.