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.

翻译：飞机空中加油问题是一个具有$n!$个可行解的非线性组合优化问题。给定一个由$n$架具备空中加油技术的飞机组成的机队，每架飞机具有特定的燃油容量和燃油消耗率。机队共同飞往同一目标，在航程中，每架飞机可瞬时为其他飞机加油并随即脱离编队。问题在于如何找到最优加油策略，使最后一架剩余飞机飞行距离最远。为在多项式时间内解决大规模飞机空中加油问题，我们利用该问题的数据结构特性，提出了序贯可行解的定义。我们证明：若飞机空中加油问题存在可行解，则必然存在序贯可行解，且其最优可行解必为最优序贯可行解。随后我们提出序贯搜索算法，其计算复杂度取决于序贯可行解的数量$Q_n$。理论上界为指数界$2^{n-2}$，在最坏情况下缺乏对大输入规模的适用性。因此，我们从动态视角研究序贯搜索算法的复杂度行为，发现当输入量$n$大于$2m$时，$Q_n$受限于$\frac{m^2}{n}C_n^m$。此处$m$为常数，$2m$被视为序贯搜索算法复杂度从指数时间向多项式时间转换的"拐点"。此外，我们构建了一个高效可计算性框架，可据此预测序贯搜索算法的具体复杂度，从而帮助决策者或用户根据可用运行时间选择适当的算法。