The airplane refueling problem is NP-complete and is solvable in polynomial time

Abstract: The airplane refueling problem is a nonlinear combinatorial optimization problem, and its equivalent problem the $n$-vehicle exploration problem is proved to be NP-complete (arXiv:2304.03965v1, The $n$-vehicle exploration problem is NP-complete). In Article (arXiv:2210.11634v2, A polynomial-time algorithm to solve the aircraft refueling problem: the sequential search algorithm), we designed the sequential search algorithm for solving large scale of airplane refueling instances, and we proved that the computational complexity increases to polynomial time with increasing number of airplanes. Thus the airplane refueling problem, as an NP-complete problem, is solvable in polynomial time when its input scale is sufficiently large.