An Improved Randomized On-Line Algorithm for a Weighted Interval Selection Problem

Given a set of weighted intervals, the objective of the weighted interval selection problem (WISP) is to select a maximum-weight subset such that the selected intervals are pairwise disjoint. We consider on-line algorithms that process the intervals in order of non-decreasing left endpoints. Preemption is allowed, but rejections are irrevocable. This problem has natural applications in various scheduling problems. We study the class of monotone instances of WISP, i.e., we require that the order of right endpoints of the given intervals coincides with that of the left endpoints. This class includes the case where all intervals have the same length. For monotone instances of WISP, the best possible competitive ratio for deterministic on-line algorithms is known to be 1/4. It has long been an open question whether there exists a randomized algorithm with better competitive ratio. In this paper, we present a new randomized algorithm and prove that it achieves a better competitive ratio 1/3 for the special case of monotone WISP where the sequence of weights of the arriving intervals is non-decreasing. Thus we provide the first result towards a solution of the long-standing open question. Furthermore, we show that no randomized algorithm achieves a competitive ratio strictly larger than 4/5. This is the first non-trivial upper bound for randomized algorithms for monotone WISP.