Rent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental

TL;DR

This paper introduces randomized algorithms for the Multislope Ski Rental problem, a generalization of the classical ski rental problem, providing optimal strategies for additive cases and an $e$-competitive approach for general instances.

Contribution

It presents the best possible randomized strategy for additive instances and an $e$-competitive algorithm for non-additive cases in the Multislope Ski Rental problem.

Findings

Optimal randomized strategy for additive instances.

An $e$-competitive randomized algorithm for general instances.

Extension of classical ski rental problem to multislope scenarios.

Abstract

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 option at any point in time. Multislope Ski Rental is a natural generalization of the classical Ski Rental problem (where the only options are 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 decisions cannot be modeled by just two options, 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 the best possible online randomized strategy for any additive instance, where the cost of switching from one option to another is the difference in their buying prices; and an algorithm that produces an $e$-competitive randomized strategy for any (non-additive) instance.