Fortune's Bounty: Taming Complexity by Trimming Trees --- A Hands-On Problem-Solving Experience in Advanced Complexity Suitable for Introductory Students

Abstract

This article provides an assignment designed to let undergraduate students who have completed an undergraduate CS1/CS2 sequence try to themselves, in groups, prove Fortune's Theorem. (Fortune's Theorem states that if the complement of the Boolean satisfiability problem polynomial-time reduces to a sparse set, then the Boolean satisfiability problem is polynomial-time computable. The assignment does not assume that students have previously seen the Boolean satisfiability problem, polynomial-time reductions, or sparse sets. Rather, it teaches those within the assignment. Note: Reworded into the technical vocabulary of complexity theory, Fortune's Theorem states that no sparse set is coNP-hard unless P=NP. Fortune's Theorem was a major advance in the understanding of the relationship between hardness and density.) We provide both the assignment handout (as the main body of this report plus Appendix A) and a solution to the assignment (as Appendix B, which would of course not be made available to the students until after they had handed in the assignment). The assignment handout, though the instructor can change this, is framed as having the students starting the assignment in teams in class for a whole class session, and then finishing it in those same teams as a take-home assignment, and handing it in before the next class session. We have found that student groups often succeed, partially or completely, in this challenge. This can mean a lot to the students: they see that they were able to make an advance that, when it was first obtained, appeared in what was arguably at the time the top journal venue for complexity theory research. This can give them confidence that they have substantial problem-solving skills (which basically means research skills) when they truly apply themselves to a given challenge.