This was the second lecture on NP-completeness. The first lecture introduced the concept, and this lecture developed a series of reductions. I decided that the first lecture would not be a good one for student submissions, since it covered a number of concepts that would have been difficult to do as activities. This lecture seemed like a better fit, since it emphasized problem transormations and I wanted the students to get a feel for individual problems. I decided not to cover some of the more challenging resutls (such as the reduction from 3-Sat to Independent set). My recollection had been that on previous occasions, students got pretty lost on the complicated reductions - so that time could be freed up for other use.
The first activity was to determine a solution to an instance of Circuit SAT. I had introduced Cook's theorem - and wanted students to have a feel for the circuit sat problem. The example had a good complexity - students were very engaged in solving the problem - every group had a lot of collaboration/discussion. Students seemed to solve the problem in about 10 minutes. Students generally worked backwards, and then checked the answer. Students were successful in finding the solution.
This problem was also to find a truth assignment - this time for a 3-SAT formula. The main goal was for students to get additional experience with the problem. The example was chosen so that it did not have a solution, and so that there was a structure/interpretation to the clauses. The collaboration level on this one was also high, and I was impressed at the number of students that reasoned about the meaning of the clauses. Almost all students determined that there was no solution to this problem.
The third planned activity was to work through a small example of vertex cover. I felt that this activity was pretty weak - so I covered it directly, as opposed to having a submission. The activity was unclear (what were students to do), and trivial. I judged (correctly) that if I spead up coverage, I would get through all of the lecture.
This activity was to derive the Independent set to Clique reduction. The complement was defined on the slide - and then students filled in the complement on the graph. This was a mechanical activity, that was then used to illustrate the correspondance between the independent set and the clique. A student example was used to illustrate that the same vertices formed the independent set in one graph, and the clique in the other.
The final example was to find a traveling salesman tour in a graph. This was just to give the students a challenge, and make the problem more interesting (since I assume that they understood the definition). The size and complexity of the problem was appropriate.
