2. [50 points] Weighted Completion Time Scheduling (from Kleinberg and A small business say, a photocopying service with a single large machine faces the following scheduling problem. Each morning, they get a set of jobs from customers. They want to do the jobs on their single machine in an order that keeps their customers the happiest. Customer i's job will take ti time to complete. Given a schedule (i.e. an ordering of the jobs), let Ci denote the finishing time of job i. For example, if job j is the first to be done, we would have Cj t; and if job j is done right after job i, we would have Cj Ci +tj. Each customer i also has a given weight w; that represents his or her importance to the business. The happiness of customer i is expected to be dependent on the finishing time of i's job So the company decides that they want to order the jobs to minimize the weighted sum of completion times 1 UG In summary: you are given a set of jobs with processing time t, and weight wi for each job. You want to order the jobs so as to minimize the weighted sum of the completion tieswC Example: Suppose their are two jobs: the first takes time t1 and has weight wi 10, while the second job takes time t23 and has weight w2-2. Then doing job 1 first would yield a weighted completion time of 10-1 +2-4-18, while doing the second job first would yield the larger weighted completion time of 10 4+2.3 46 (a) [10 points] Propose a plausible greedy algorithm for this problem and prove that it does not always yield an optinm (b) [40 points] Describe and analyze an efficient algorithm to solve this