A Jeep Crossing a Desert of Unknown Width

Abstract

The classic jeep problem concerns crossing a desert wider than the range of the jeep, with the aid of preplaced fuel caches. There has been a lot of work on this problem and its variations, and the optimal strategy is well known, but all previous work assumes that we know the width of the desert. We consider the case where we don’t know the distance in advance. We evaluate a strategy by its competitive ratio, which is the worst-case ratio of the cost of the strategy, divided by the cost of an optimal solution had we known the distance in advance. We show that no strategy with a fixed sequence of caches can achieve a finite competitive ratio. The optimal strategy is an iterative one that uses the optimal known-distance strategy to reach a sequence of target distances, emptying all caches between iterations. An optimal iterative strategy doubles the cost of each successive iteration, and achieves a competitive ratio of four.