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.

我们考虑以下经典滑雪租赁问题的概括。持续时间未知的任务必须使用“购买”和“租赁”两种选择之一来执行，每种选择都有一次性启动成本和持续成本，该成本是持续时间的函数。从租赁转为购买也会产生一次性成本。目标是最小化竞争比，即在所有可能的持续时间内支付的成本与最优成本之间的最坏情况比率。对于线性或指数成本函数，最佳的确定性和随机在线策略是众所周知的。在这项工作中，我们分析了一个更一般的情况，仅假设成本函数是连续的并且满足某些温和的单调性条件。对于这种一般情况，我们提供了一种算法来计算具有最佳竞争比的确定性策略，\(\epsilon >0\) ，根据\( \epsilon ^{-1}\)中的时间多项式计算最佳可能的竞争比在 \((1+\epsilon )\) 范围内的随机策略。我们的算法假设可以访问一个黑匣子，该黑匣子可以计算函数及其反函数，并找到它们的极值点。