This lecture covers a very elegant application of network flow: the use of the minimum cut to find an optimal selection of tasks. This results shows the tremendous power of network flow to address problems that look nothing like flows passing through pipes. The goal of this lecture is to cover this one particular construction in detail, so that students can apply networkflow to this type of selection problem. Although the construction is fairly easy to state - there are a number of subtle points - so the lecture covers these in detail. The design of the lecture is to support a process of discovery - examples are used to show the particular technical points before they are stated and proved.
An initial analysis of student submissions was started before class took place!
Activity Goals: Students are asked to solve a fairly big example of the problem by hand. The goal is to get students thinking about the problem, and to have some vested interest in it's solution. This activy should also ensure that students understant what the problem is that is being solved.
Planned Use: Show a single solution with the correct answer.
Actual use: Several similar correct solutions were displayed (just to put up a number of students work). A final submission was shown which had a comment - which allowed a small joke.
Evaluation of activity: The activity was successful in engaging the students. The particular example was not that well designed - there was an 'obvious solution'. More thought should have been put into designing the example to make it a little more challenging.
Activity Goals: There are two goals for this activity. First, to verify that students understand the role of the infinite capacity edges, and that the edges can go from T to S, but not from S to T. Second, to discover that the set T is a feasible set.
Planned Use: Show multiple correct solutions to show that all of the correct solutions are feasible. This is counting on having a variety of correct solutions.
Actual use:Initial submissions (not displayed) showed that many students did not understand how to find a finite cut - so students were encouraged to talk with one another, and various clarifications were made. Some students revised their solutions. When the activity was finished I showed a number of solutions - including a few incorrect solutions.
Evaluation of activity: I thought the activity worked well for clarifying the first part of the instruction. The display of multiple correct solutions might not have conveyed that the solutions were all "feasible".
Activity Goals: This activity is to have students generate all finite cuts and compute the cut's value. Hopefully, students will see that the cut with the minimum value corresponds to the optimal selection of tasks. Working this activity might also help develop intuition about what the cut means. Planned Use: Show a correct solution.
Actual use: A couple of correct solutions were displayed.
Evaluation of activity: Students generally found the desired solutions, and the correct values for the cuts. This example did seem to make the point that the cut coincided with the maximum profit.
Activity Goals: Derive an expression for the value of the cut. Once this expression is derived - it is immediate that the minimum cut maximizes the profit. I am hoping that the problem is constrained enough that some students will be able to determine the desired formula.
Planned Use: Show a correct answer, and then give the derivation on a clean slide. This activity may be hard - but even if no one gets it right - it is worth having them think about it.
Actual use:
Time was running short - so I added a hint (which made it much easier). I only waited for a few submissions before putting up a correct answer. There was a perfect double submission - which gave the answer, and then improved it. Evaluation of activity: With the shortness of time, I was happy how this worked - students thought about the answer, and I was then able to use a student provided solution.
