Optimal Placement of Base Stations in Border Surveillance using Limited Capacity Drones

TL;DR

This paper addresses the problem of optimally placing refueling stations for border surveillance drones around an island modeled as a polygon, providing algorithms for station placement and fuel capacity approximation with theoretical guarantees.

Contribution

It introduces algorithms for near-optimal station placement and fuel capacity approximation, along with a practical heuristic for certifying optimality.

Findings

Algorithm finds station locations within one of the optimal plus one.

Provides an additive epsilon-approximation for fuel capacity minimization.

Proposes a heuristic that can certify optimality under certain conditions.

Abstract

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.