A polynomial-time algorithm to solve the large scale of airplane refueling problem

Airplane refueling problem is a nonlinear combinatorial optimization problem with $n!$ feasible feasible solutions. Given a fleet of $n$ airplanes with mid-air refueling technique, each airplane has a specific fuel capacity and fuel consumption rate. The fleet starts to fly together to a same target and during the trip each airplane could instantaneously refuel to other airplanes and then be dropped out. The question is how to find the best refueling policy to make the last remaining airplane travels the farthest. To solve the large scale of the airplane refueling problem in polynomial-time, we propose the definition of the sequential feasible solution by employing the data structural properties of the airplane refueling problem. We prove that if an airplane refueling problem has feasible solutions, it must have sequential feasible solutions, and its optimal feasible solution must be the optimal sequential feasible solution. Then we present the sequential search algorithm which has a computational complexity that depends on the number of sequential feasible solutions referred to $Q_n$, which is proved to be upper bounded by $2^{n-2}$ as an exponential bound that lacks of applicability on larger input for worst case. Therefore we investigate the complexity behavior of the sequential search algorithm from dynamic perspective, and find out that $Q_n$ is bounded by $\frac{m^2}{n}C_n^m$ when the input $n$ is greater than $2m$. Here $m$ is a constant and $2m$ is regarded as the "inflection point" of the complexity of the sequential search algorithm from exponential-time to polynomial-time. Moreover, we build an efficient computability scheme according to which we shall predict the specific complexity of the sequential search algorithm to choose a proper algorithm considering the available running time for decision makers or users.