A Polynomial-time Algorithm to Solve the Airplane Refueling Problem: the Sequential Search Algorithm

TL;DR

This paper introduces a polynomial-time algorithm for the airplane refueling problem, leveraging the concept of sequential feasible solutions to efficiently find the optimal refueling policy.

Contribution

The paper defines sequential feasible solutions and proves the optimal solution is among them, leading to a polynomial-time sequential search algorithm for the problem.

Findings

The number of sequential feasible solutions grows polynomially after a certain input size.

The sequential search algorithm efficiently finds the optimal solution in polynomial time.

A computability scheme predicts the algorithm's complexity for any given instance.

Abstract

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.