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Airplane refueling problem is a nonlinear combinatorial optimization problemwith $n!$ feasible feasible solutions. Given a fleet of $n$ airplanes withmid-air refueling technique, each airplane has a specific fuel capacity andfuel consumption rate. The fleet starts to fly together to a same target andduring the trip each airplane could instantaneously refuel to other airplanesand then be dropped out. The question is how to find the best refueling policyto make the last remaining airplane travels the farthest. To solve the largescale of the airplane refueling problem in polynomial-time, we propose thedefinition of the sequential feasible solution by employing the data structuralproperties of the airplane refueling problem. We prove that if an airplanerefueling problem has feasible solutions, it must have sequential feasiblesolutions, and its optimal feasible solution must be the optimal sequentialfeasible solution. Then we present the sequential search algorithm which has acomputational complexity that depends on the number of sequential feasiblesolutions 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 worstcase. Therefore we investigate the complexity behavior of the sequential searchalgorithm 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 aconstant and $2m$ is regarded as the "inflection point" of the complexity ofthe sequential search algorithm from exponential-time to polynomial-time.Moreover, we build an efficient computability scheme according to which weshall predict the specific complexity of the sequential search algorithm tochoose a proper algorithm considering the available running time for decisionmakers or users.

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