On page 46-47, Garey&Johnson list out 6 “basic core” NP-Complete problems. I’ll go over all of them quickly, because they will be used in the reductions for many other problems. But they all start with the “first” NP-Complete problem, Satisfiabiity:
The Problem: Satisfiability (SAT)
The Description: (G&J, p. 39) Given a set U of variables, and a set C of Clauses over U, is there a way to set the variables in U such that every clause in C is true?
(Note that this is not general satisfiability, where the input is any boolean formula. This is really “CNF-Satisfiability”, where the formula is in conjunctive normal form. But you can use logical rules to turn any general logical formula into a CNF formula)
The proof: (p. 39-44) G&J provide a proof of Cook’s Theorem, which provides a “first” NP-Complete problem by representing the formula in non-deterministic Turing Machines. It’s a crazy hard proof, we spent a week (at least) of my graduate school Theory of Computation class going over it, and I still have trouble following it. I think it’s way too hard for undergraduates to follow.