# Tag Archives: sat

## 3-Satisfiability

The first of the “core six” problems, and the first NP-Complete problem that’s actually a reduction from a known NP-Complete problem.

The Problem: 3-Satisfiability (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 “CNF-SAT”, most people would call this problem “3-CNF-SAT”)

The Reduction: From SAT.  Pages 48-49 give a simple algorithm to convert arbitrarily-sized 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 3-CNF-SAT is NP-Complete, the similar problem of 2-CNF-SAT (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 3-CNF to 2-CNF, or  Fractional Knapsack to 0/1 Knapsack-where the solution space shrinks!) can drastically affect the complexity of the problem.

## Satisfiability

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 to hard for undergraduates to follow.

Difficulty: 10