Abstract
We consider the problem of scheduling a set of equal-length intervals arriving online, where each interval is associated with a weight and the objective is to maximize the total weight of completed intervals. An optimal 4-competitive algorithm has long been known in the deterministic case, but the randomized case remains open. We give the first randomized algorithm for this problem, achieving a competitive ratio of 3.5822. We also prove a randomized lower bound of 4/3, which is an improvement over the previous 5/4 result. Then we show that the techniques can be carried to the deterministic multiprocessor case, giving a 3.5822-competitive 2-processor algorithm, and a 4/3 lower bound for any number of processors. We also give a lower bound of 2 for the case of two processors.
Highlights
We study the problem of scheduling a set of intervals which arrive online
The objective is to schedule a set of non-overlapping intervals such that the total weight of all these intervals are maximized
Intervals being processed can be interrupted, but the value will be lost. This can be viewed as a job scheduling problem where each job must be served immediately or else it is lost
Summary
We study the problem of scheduling a set of intervals which arrive online. Each interval has a weight and all intervals are of the same length. For the basic problem where intervals are of the same length and with arbitrary weights, Woeginger [15] gave an optimal deterministic 4competitive algorithm and a matching lower bound. There are matching upper and lower bounds Θ(Δ/ log Δ) [14, 16] for the deterministic case These bounds apply to the interval scheduling problem. If the weight of an interval is equal to its length, the nonpreemptive case was considered in [11] They gave a randomized O((log Δ)1+ )-competitive algorithm and a Ω(log Δ) lower bound. Multiprocessor scheduling of intervals were studied in [6], giving an optimal (1-competitive) algorithm when all intervals have unit weight (and not necessarily equal length). Due to space limitations some proofs are omitted and can be found in the full version of the paper
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