TY - GEN

T1 - Non-Linear Ski Rental

AU - Patt-Shamir, Boaz

AU - Yadai, Evyatar

N1 - Publisher Copyright:
© 2020 ACM.

PY - 2020/7/6

Y1 - 2020/7/6

N2 - 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 strategies 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 (1) an algorithm that computes the deterministic strategy with the best competitive ratio, and (2) an approximation algorithm that, given ϵ>0$, computes a randomized strategy whose competitive ratio is within (1+ϵ) from the best possible, in time polynomial in ϵ-1. Our algorithm assumes access to a black box that can compute the functions and their inverses, as well as find their extreme points.

AB - 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 strategies 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 (1) an algorithm that computes the deterministic strategy with the best competitive ratio, and (2) an approximation algorithm that, given ϵ>0$, computes a randomized strategy whose competitive ratio is within (1+ϵ) from the best possible, in time polynomial in ϵ-1. Our algorithm assumes access to a black box that can compute the functions and their inverses, as well as find their extreme points.

UR - http://www.scopus.com/inward/record.url?scp=85088630851&partnerID=8YFLogxK

U2 - 10.1145/3350755.3400280

DO - 10.1145/3350755.3400280

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AN - SCOPUS:85088630851

T3 - Annual ACM Symposium on Parallelism in Algorithms and Architectures

SP - 431

EP - 440

BT - SPAA 2020 - Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures

PB - Association for Computing Machinery

Y2 - 15 July 2020 through 17 July 2020

ER -