  Imagine an island modeled as a simple polygon $\P$ with $n$ vertices whose
coastline we wish to monitor. We consider the problem of building the minimum
number of refueling stations along the boundary of $\P$ in such a way that a
drone can follow a polygonal route enclosing the island without running out of
fuel. A drone can fly a maximum distance $d$ between consecutive stations and
is restricted to move either along the boundary of $\P$ or its exterior (i.e.,
over water). We present an algorithm that, given $\mathcal P$, finds the
locations for a set of refueling stations whose cardinality is at most the
optimal plus one. The time complexity of this algorithm is $O(n^2 + \frac{L}{d}
n)$, where $L$ is the length of $\mathcal P$. We also present an algorithm that
returns an additive $\epsilon$-approximation for the problem of minimizing the
fuel capacity required for the drones when we are allowed to place $k$ base
stations around the boundary of the island; this algorithm also finds the
locations of these refueling stations. Finally, we propose a practical
discretization heuristic which, under certain conditions, can be used to
certify optimality of the results.
