A polynomial-time approximation scheme for the airplane refueling problem

We consider the airplane refueling problem that was introduced by Gamow and Stern in their classical book Puzzle-Math (1958). In this setting, we wish to deliver a bomb the farthest possible distance, being much greater than the range of any individual airplane at our disposal. For this purpose, the only feasible option is to better utilize our fleet via mid-air refueling. Starting with a fleet of airplanes that can instantaneously refuel one another and gradually drop out of formation, how would we design the best refueling policy, i.e., one that maximizes the distance traveled by the last remaining plane? Even though Gamow and Stern provided an elegant characterization of the optimal refueling policy for the special case of identical airplanes, the general problem with arbitrary tank volumes and consumption rates has remained widely open, as pointed out by Woeginger (Albers et al., Dagstuhl seminar proceedings 10071, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany, 2010). To our knowledge, other than a logarithmic approximation, which can be attributed to folklore, improved performance guarantees have not been obtained to date. In this paper, we propose a polynomial-time approximation scheme for the airplane refueling problem in its utmost generality. Our approach employs widely-known techniques related to geometric rounding, time stretching, guessing arguments, and timeline partitions. These are augmented by additional insight and ideas, that enable us to devise reductions to well-structured instances of generalized assignment and to exploit LP-rounding algorithms for the latter problem. We complement this result by presenting a fast and easy-to-implement algorithm that attains a constant factor approximation for the optimal refueling policy.