The n-vehicle exploration problem is NP-complete

The $n$-vehicle exploration problem (NVEP) is a nonlinear unconstrained optimization problem. Given a fleet of $n$ vehicles with mid-trip refueling technique, the NVEP tries to find a sequence of $n$ vehicles to make one of the vehicles travel the farthest, and at last all the vehicles return to the start point. NVEP has a fractional form of objective function, and its computational complexity of general case remains open. Given a directed graph $G$, it can be reduced in polynomial time to an instance of NVEP. We prove that the graph $G$ has a hamiltonian path if and only if the reduced NVEP instance has a feasible sequence of length at least $n$. Therefore we show that Hamiltonian path $\leq_P$ NVEP, and consequently prove that NVEP is NP-complete.