Optimal Placement of Base Stations in Border Surveillance using Limited Capacity Drones
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Imagine an island modeled as a simple polygon 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 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 or its exterior (i.e., over water). We present an algorithm that, given 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 + L/d n), where L is the length of 𝒫. We also present an algorithm that returns an additive ϵ-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.
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