On the Best Possible Competitive Ratio for Multislope Ski Rental

The multislope ski-rental problem [LPSR08a] is an extension of the classical ski-rental problem [KMRS88], where the player has several lease options in addition to the pure rent and buy options. In this problem an instance, which is the setting of the options, significantly affects the player’s performance. There is an algorithm that for a given instance, computes the best possible strategy [AIS08]. However, the output is given in numerical values and the relational nature between the instance and the best possible performance has not been known. In this paper we prove that even for the easiest instance, a competitive ratio smaller than e/(e − 1) ≈ 1.58 cannot be achieved. More precisely, according to the number of options, tight bounds are obtained in a closed form. Furthermore, we establish a matching upper and lower bound of the competitive ratio each for the 3-slope and 4-slope problems.