A Better bound of randomized algorithms for the multislope ski-rental problem^∗

¹ School of Mathematical Science, Huaiyin Normal University, Huaian, Jiangsu 223300, P.R. China.
humaolin2000@163.com
² School of Business Administration, South China University of Technology, Guangzhou 510641, P.R. China.
xuwj@scut.edu.cn

Received: 11 April 2017
Accepted: 11 September 2017

Abstract

The multislope ski-rental problem is an extension of the classical ski-rental problem, where the player has several lease options in addition to the pure rent and buy options. For the additive general model, Lotker, Patt-Shamir and Rawitz [in: SIAM J. Discr. Math. 26 (2012) 718–736] obtained a randomized algorithm with the competitive ratio bounded by $\frac{\mathrm{e}\mathrm{-}{\mathit{r}}_{\mathit{k}}\mathit{/}{\mathit{r}}_{\mathrm{0}}}{\mathrm{e}\mathrm{-}\mathrm{1}}$. However, obtaining a better bound on the competitive factor as a function of the slopes parameters remains an open problem in their paper. In this paper, we study randomized algorithm for the additive multislope ski rental problem, and extend the competitive ratio bound $\frac{\mathrm{e}\mathrm{-}{\mathit{r}}_{\mathit{k}}\mathit{/}{\mathit{r}}_{\mathrm{0}}}{\mathrm{e}\mathrm{-}\mathrm{1}}$ proposed by Lotker et al. to $\frac{\mathrm{e}}{\mathrm{e}\mathrm{-}\mathrm{1}\mathrm{+}{\mathit{r}}_{\mathit{k}}\mathit{/}{\mathit{r}}_{\mathrm{0}}}$.