Dynamic Interval Scheduling for Multiple Machines

Abstract

We study the dynamic scheduling problem for jobs with fixed start and end times on multiple machines. The problem is to maintain an optimal schedule under the update operations: insertions and deletions of jobs. Call the period of time in a schedule between two consecutive jobs in a given machine an idle interval. We show that for any set of jobs there exists a schedule such that the corresponding set of idle intervals forms a tree under the set-theoretic inclusion. Based on this result, we provide a data structure that updates the optimal schedule in $O(d+\log n)$ worst-case time, where $d$ is the depth of the set idle intervals and $n$ is the number of jobs. Furthermore, we show this bound to be tight for any data structure that maintains a nested schedule.