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

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 \hbox{$\frac{{\rm e}-r_{k}/r_{0}}{{\rm e}-1}$}. 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 \hbox{$\frac{{\rm e}-r_{k}/r_{0}}{{\rm e}-1}$} proposed by Lotker et al. to \hbox{$\frac{{\rm e}}{{\rm e}-1+r_{k}/r_{0}}$}