Some of your friends have recently graduated and started a small company, which they are currently running out of their parents' garages in Santa Clara. They're in the process of porting all their software from an old system to a new, revved-up system; and they're facing the following problem. They have a collection of n software applications, {1, 2,...., n}, running on their old system; and they'd like to port some of these to the new system. If they move application i to the new system, they expect a net (monetary) benefit of bi >= 0. The different software applications interact with one another; if applications i and j have extensive interaction, then the company will incur an expense if they move one of i or j to the new system but not both; let's denote this expense by xij >= 0.
So, if the situation were really this simple, your friend would just port all n applications, achieving a total benefit of Si bi. Unfortunately, there's a problem. Due to small but fundamental incompatibilities between the two systems, there's no way to port application 1 to the new system; it will have to remain on the old system. Nevertheless, it might still pay off to port some of the other applications, accruing the associated benefit and incurring the expense of the interaction between applications on different systems.
So this is the question they pose to you: Which of the remaining applications, if any, should be moved? Give a polynomial-time algorithm to find a subset S of {2, 3,..., n} for which the sum of the benefits minus expenses of moving the applications in S to the new system is maximized.