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We present a fast algorithm for the following classic scheduling problem: Determine a maximum-weight schedule for a collection of unit jobs, each of which has an associated release time, deadline, and weight. All previous algorithms for this problem have at least quadratic worst-case complexity. This job scheduling problem can also be viewed as a special case of weighted bipartite matching: each job represents a vertex on the left side of the bipartite graph; each time slot represents a vertex on the right side; each job is connected by an edge to all time slots between its release time and deadline; all of the edges adjacent to a given job have weight equal to the weight of the job. Letting U denote the set of jobs and V denote the set of time slots, our algorithm runs in O(|U| + klog2 k) time, where k ≤ min {|U|,|V|} denotes the cardinality of a maximum-cardinality matching. Thus our algorithm runs in nearly linear time, a dramatic improvement over the previous quadratic bounds.

Fast Scheduling of Weighted Unit Jobs with Release Times and Deadlines