Recall the scheduling problem from Section in which we sought to minimize the maximum late...

Recall the scheduling problem from Section in which we sought to minimize the maximum lateness. There are n jobs, each with a deadline di and a required processing time ti, and all jobs are available to be scheduled starting at time s. For a job i to be done, it needs to be assigned a period from > s to fi = si + ti, and different jobs should be assigned nonoverlapping intervals. As usual, such an assignment of times will be called a schedule.

In this problem, we consider the same setup, but want to optimize a different objective. In particular, we consider the case in which each job must either be done by its deadline or not at all. We'll say that a subset J of the jobs is schedulable if there is a schedule for the jobs in J so that each of them finishes by its deadline. Your problem is to select a schedulable subset of maximum possible size and give a schedule for this subset that allows each job to finish by its deadline.

(a) Prove that there is an optimal solution J (i.e., a schedulable set of maximum size) in which the jobs in J are scheduled in increasing order of their deadlines.

(b) Assume that all deadlines di and required times ti are integers. Give an algorithm to find an optimal solution. Your algorithm should run in time polynomial in the number of jobs n, and the maximum deadline D = maxi di.