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3Satisfiability
The first of the “core six” problems, and the first NPComplete problem that’s actually a reduction from a known NPComplete problem.
The Problem: 3Satisfiability (3SAT)
The Definition: (p.46) Given a set of variables U, and a set C of clauses, where each clause has exactly 3 elements, can we assign true/false values to the variables in U that satisfies all of the clauses in C?
(Note that just as the Satisfiability problem was really “CNFSAT”, most people would call this problem “3CNFSAT”)
The Reduction: From SAT. Pages 4849 give a simple algorithm to convert arbitrarilysized clauses into clauses of size 3, possibly by adding some extra variables.
Difficulty: 6. It’s pretty fiddly to come up with the conversions in all cases. But this really depends on the students’ familiarity with Boolean logic.
As an aside, I find myself compelled to note that while 3CNFSAT is NPComplete, the similar problem of 2CNFSAT (where all clauses are of size 2) can be solved in polynomial time. To me, one of the coolest things in Computer Science is how such small changes to a problem (like 3CNF to 2CNF, or Fractional Knapsack to 0/1 Knapsackwhere the solution space shrinks!) can drastically affect the complexity of the problem.
Satisfiability
On page 4647, Garey&Johnson list out 6 “basic core” NPComplete 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” NPComplete 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 “CNFSatisfiability”, 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. 3944) G&J provide a proof of Cook’s Theorem, which provides a “first” NPComplete problem by representing the formula in nondeterministic 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 to hard for undergraduates to follow.
Difficulty: 10