Airplane refueling problem is a nonlinear unconstrained optimization problem with $n!$ feasible solutions. Given a fleet of $n$ airplanes with mid-air refueling technique, the question is to find the best refueling policy to make the last remaining airplane travel the farthest. In order to solve airplane refueling problem, we proposed the definition of sequential feasible solution by employing the refueling properties of data structure. We proved that if an airplane refueling instance has feasible solutions, it must have sequential feasible solutions; and the optimal feasible solution must be the optimal sequential feasible solution. So we need to numerate all the sequential feasible solutions to get an exact algorithm. We proposed the sequential search algorithm which consists of two steps, the first step of which aims to seek out all of the sequential feasible solutions, and the second step aims to search for the maximal sequential feasible solution by bubble sorting all of the sequential feasible solutions. We observed that the number of the sequential feasible solutions will change to grow at a polynomial rate when the input size of $n$ is greater than an inflection point $N$. Then we proved that the sequential search algorithm is a polynomial-time algorithm to solve the airplane refueling problem. Moreover, we built an efficient computability scheme, according to which we could forecast within a polynomial time the computational complexity of the sequential search algorithm that runs on any given airplane refueling instance. Thus we could provide a computational strategy for decision makers or algorithm users by considering with their available computing resources.

翻译：飞机空中加油问题是一个具有$n!$个可行解的非线性无约束优化问题。给定一支由$n架具备空中加油技术的飞机组成的机队，问题在于寻找最优的加油策略，使得最后一架剩余飞机的飞行距离最远。为解决飞机空中加油问题，我们利用数据结构的加油性质提出了顺序可行解的定义。我们证明了：若一个飞机加油实例存在可行解，则其必然存在顺序可行解；且最优可行解必为最优顺序可行解。因此，我们需要枚举所有顺序可行解以获得精确算法。我们提出了顺序搜索算法，该算法包含两个步骤：第一步旨在找出所有顺序可行解，第二步则通过冒泡排序对所有顺序可行解进行排序以搜寻最大顺序可行解。我们观察到，当输入规模$n$大于拐点$N$时，顺序可行解的数量将转变为以多项式速率增长。随后我们证明了顺序搜索算法是求解飞机空中加油问题的多项式时间算法。此外，我们构建了一个高效可计算性方案，依据该方案我们能够在多项式时间内预测顺序搜索算法在任何给定飞机加油实例上运行时的计算复杂度。从而，决策者或算法使用者可根据其可用计算资源，获得相应的计算策略建议。