Rent, lease, or buy: Randomized algorithms for multislope ski rental

In the multislope ski rental problem, the user needs a certain resource for some unknown period of time. To use the resource, the user must subscribe to one of several options, each of which consists of a one-time setup cost ("buying price") and cost proportional to the duration of the usage ("rental rate"). The larger the price, the smaller the rent. The actual usage time is determined by an adversary, and the goal of an algorithm is to minimize the cost by choosing the best alternative at any point in time. Multislope ski rental is a natural generalization of the classical ski rental problem (where there are only two available alternatives, namely pure rent and pure buy), which is one of the fundamental problems of online computation. The multislope ski rental problem is an abstraction of many problems, where online choices cannot be modeled by just two alternatives, e.g., power management in systems which can be shut down in parts. In this paper we study randomized algorithms for multislope ski rental. Our results include an algorithm that produces the best possible online randomized strategy for any additive instance, where the cost of switching from one alternative to another is the difference in their buying prices, and an e-competitive randomized strategy for any (not necessarily additive) instance.