Online interval scheduling

We introduce the online interval scheduling problem, in which a set of intervals of the positive real line is presented to a scheduling algorithm in order of start time. Upon seeing each interval, the algorithm must decide whether or not to “schedule” it. Overlapping intervals may not be scheduled together. We give a strongly 2-competitive algorithm for the case in which intervals must be one of two lengths, either length 1 or length k >> 1. For the general case in which intervals may have arbitrary lengths, A, the ratio of longest to shortest interval, is the important parameter. We give an algorithm with competitive factor O((log A)l+E), and show that no O(logA)-competitive algorithm can exist. Our algorithm need not know the ratio A in advance.