The Oil Searching Problem

Abstract

Given n potential oil locations, where each has oil at a certain depth, we seek good trade-offs between the number of oil sources found and the total amount of drilling performed. The cost of exploring a location is proportional to the depth to which it is drilled. The algorithm has no clue about the depths of the oil sources at the different locations or even if there are any. Abstraction of the oil searching problem applies naturally to several life contexts. Consider a researcher who wants to decide which research problems to invest time into. A natural dilemma whether to invest all the time into a few problems, or share time across many problems. When you have spent a lot of time on one problem with no success, should you continue or move to another problem?

One could study this problem using a competitive analysis that compares the cost of an algorithm to that of an adversary that knows the depths of the oil sources, but the competitive ratio of the best algorithm for this problem is Ω(n). Instead we measure the performance of a strategy by comparing it to a weaker adversary that knows the set of depths of the oil sources but does not know which location has what depth. Surprisingly, we find that to find k oil sources there is a strategy whose cost is close to that of any adversary that has this limited knowledge of only the set of depths. In particular, we show that if any adversary can find k oil sources with drilling cost B while knowing the set of depths, our strategy finds $k - \tilde O(k^{5/6})$ sources with drilling cost B(1 + o(1)). This proves that our strategy is very close to the best possible strategy in the total absence of information.