We study the optimal solution to a general two-slope ski rental problem with a tail risk, i.e., the chance of the competitive ratio exceeding a value $\gamma$ is bounded by $\delta$. This extends the recent study of tail bounds for ski rental by [Dinitz et al. SODA 2024] to the two-slope version defined by [Lotker et al. IPL 2008]. In this version, even after "buying" we must still pay a rental cost at each time step, though it is lower after buying. This models many real-world "rent-or-buy" scenarios where a one-time investment decreases (but does not eliminate) the per-time cost. Despite this being a simple extension of the classical problem, we find that adding tail risk bounds creates a fundamentally different solution structure. For example, in our setting there is a possibility that we never buy in an optimal solution (which can also occur without tail bounds), but more strangely (and unlike the case without tail bounds or the classical case with tail bounds) we also show that the optimal solution might need to have nontrivial probabilities of buying even at finite points beyond the time corresponding to the buying cost. Moreover, in many regimes there does not exist a unique optimal solution. As our first contribution, we develop a series of structure theorems to characterize some features of optimal solutions. The complex structure of optimal solutions makes it more difficult to develop an algorithm to compute such a solution. As our second contribution, we utilize our structure theorems to design two algorithms: one based on a greedy algorithm combined with binary search that is fast but yields arbitrarily close to optimal solutions, and a slower algorithm based on linear programming which computes exact optimal solutions.

翻译：本文研究具有尾部风险的一般双斜率滑雪租赁问题的最优解，即竞争比超过某一值$\gamma$的概率以$\delta$为界。这扩展了[Dinitz等人，SODA 2024]近期对滑雪租赁问题尾部界的研究，将其推广至[Lotker等人，IPL 2008]定义的双斜率版本。在此版本中，即使在"购买"后，每个时间步仍需支付租赁成本（尽管购买后成本更低）。这模拟了许多现实世界"租或买"场景，其中一次性投资会降低（但不会消除）单位时间成本。尽管这是经典问题的简单扩展，但我们发现添加尾部风险界会产生根本不同的解结构。例如，在我们的设定中，最优解可能存在永不购买的可能性（这在无尾部界的情况下也可能发生），但更奇特的是（与无尾部界的情况或具有尾部界的经典情况不同），我们还证明最优解可能需要在超过购买成本对应时间后的有限时间点以非平凡概率进行购买。此外，在许多参数区间中并不存在唯一的最优解。作为我们的第一个贡献，我们建立了一系列结构定理来刻画最优解的某些特征。最优解的复杂结构使得设计计算此类解的算法更加困难。作为我们的第二个贡献，我们利用结构定理设计了两种算法：一种是基于贪心算法结合二分搜索的快速算法，可产生任意接近最优的解；另一种是基于线性规划的较慢算法，可计算精确的最优解。