Non-Linear Ski Rental

We consider the following generalization of the classic ski rental problem. A task of unknown duration must be carried out using one of two alternatives called “buy” and “rent”, each with a one-time startup cost and an ongoing cost which is a function of the duration. Switching from rent to buy also incurs a one-time cost. The goal is to minimize the competitive ratio, i.e., the worst-case ratio between the cost paid and the optimal cost, over all possible durations. For linear or exponential cost functions, the best deterministic and randomized on-line trategies are well known. In this work we analyze a much more general case, assuming only that the cost functions are continuous and satisfy certain mild monotonicity conditions. For this general case we provide an algorithm that computes the deterministic strategy with the best competitive ratio, and an algorithm that, given \(\epsilon >0\), computes a randomized strategy whose competitive ratio is within \((1+\epsilon )\) from the best possible, in time polynomial in \(\epsilon ^{-1}\). Our algorithm assumes access to a black box that can compute the functions and their inverses, as well as find their extreme points.