Consider a case where there are n tasks, with time requirements r$1,$ r$2,...,$ rn hours. On the project team, there are k people with time availabilities a$1,$ a$2,...,$ ak$.$ For each task i and person j$,$ you are told if person j has the skills to do task i$.$ You are to decide if the tasks can be split up among the people so that all tasks get done. People only execute tasks they are qualified for, and no one exceeds his time availability. Remember that you can split up one task between multiple qualified people. Now, in addition, there are group constraints. For instance, even if each of you and your two teammates can in principle spend $4$ hours on the project, you may have decided that between the three of you, you only want to spend $10$ hours. Formally, we assume that there are m sets Sj $\{1,...,$ k$\}$ of people, with set constraints tj $.$ Then, any valid solution must ensure, in addition to the previous constraints, that the combined work of all people in Sj does not exceed tj $,$ for all j$.$ Note that one person may belong to at most one constraint set. Give an algorithm with running time polynomial in n$,$ m$,$ k for this problem, under the assumption that all the Sj are disjoint, and prove that your algorithm is correct.