NP-Completeness - Hampden-Sydney College

NP- 

Completeness 

Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Assignment 

NP-Completeness 

Lecture 40 

Section 7.4 

Robb T. Koether 

Hampden-Sydney College 

Wed, Dec 3, 2008

Outline 



Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Assignment 

1 Homework Review 

2 NP-Completeness 

3 The SAT Problem 

4 Cook’s Theorem 

5 The Reduction of 3SAT to CLIQUE 

6 Assignment

Homework Review 



Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Assignment 

Exercise 7.12, page 295. 

Let 

MODEXP = {〈a, b, c, p〉 | a, b, c, and p are binary 

integers such that a b ≡ c (mod p)}. 

Show that MODEXP ∈ P. (Note that the most obvious 

algorithm doesn’t run in polynomial time. Hint: Try it first 

where b is a power of 2.)




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Assignment 

Solution 

The efficient algorithm to raise a number to the power n 

(n > 0) is to use repeated squaring. 

For example, 

a 16 = 

( ((a 

2 ) 2 ) 2 ) 2 

. 

If the exponent is not a power of 2, then there are some 

more details. 

For example, 

a 13 = 

( (a2 · a ) 2 ) 2 

· a.




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Assignment 

Solution 

Use this algorithm together with reduction modulo p 

after each operation. 

For example, 3 13 (mod 5): 

3 2 ≡ 9 ≡ 4 (mod 5). 

4 · 3 ≡ 12 ≡ 2 (mod 5). 

2 2 ≡ 4 (mod 5). 

4 2 ≡ 16 ≡ 1 (mod 5). 

1 · 3 ≡ 3 (mod 5).




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Factoring 

On a related note, if it were possible (is possible) to 

compute (n − 1)! (mod n) in polynomial time, then it 

would be possible (is possible) to factor n in polynomial 

time. 

How so? 

Assignment

NP-Completeness 



Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Certain problems in NP have been shown to have a 

special property: 

If they are solvable deterministically in polynomial time, 

then so are all problems in NP. 

These problems are called NP-complete. 

We will show that SAT is NP-complete. 

Assignment

The Satisfiability Problem 



Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

The SAT Problem 

Given a Boolean expression in m variables x 1 , x 2 , . . . , x m 

consisting of n literals (occurrences of x i or ¬x i ), do there 

exist Boolean values for x 1 , x 2 , . . . , x m that make the 

expression true? (n is the size of the problem.) 

Assignment

The Satisfiability Problem 



Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

For example, is the expression 

(x ∧ (y ∨ ¬z)) ∧ ((¬x ∨ y) ∧ z) ∨ ¬ (x ∧ ¬z) 

satisfiable? 

How can we tell? 

Assignment

Cook’s Theorem 



Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Assignment 

Theorem (Cook’s Theorem) 

SAT ∈ P if and only if P = NP. 

Proof idea. 

First show that SAT is in NP. 

Then show that every problem in NP is polynomial-time 

reducible to SAT. 

Details in the next lecture.

Polynomial-Time Computability 



Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Assignment 

Definition (Polynomial-time computable) 

A function f : Σ ∗ → Σ ∗ is polynomial-time computable if 

there is a polynomial-time Turing machine M that halts on 

all inputs w with f(w) on its tape. 

Definition (Polynomial-time reducible) 

A language A is polynomial-time reducible to a language B 

if there exists a polynomial-time computable function f such 

that 

w ∈ A ⇔ f(w) ∈ B. 

We write A ≤ P B.

NP-Complete 



Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

Definition 

A language B is NP-complete if B ∈ NP, and every problem 

A in NP is polynomial-time reducible to B. 

The 

Reduction of 

3SAT to 

CLIQUE 

Assignment

A Polynomial-Time Reduction 



Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Before proving Cook’s Theorem, we will look at an 

example of a polynomial-time reduction. 

We will reduce 3SAT to CLIQUE. 

That is, we will show that 

3SAT ≤ P SAT. 

Assignment

Reduction of 3SAT to CLIQUE 



Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Let f be a Boolean expression in conjunctive normal 

form with exactly 3 literals per clause. 

Create a graph G by the following two steps. 

(1) For each clause, create a group of nodes labeled with 

the literals in that clause. 

Assignment




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

For example, let 

f = (x∨y ∨z)∧(¬x∨¬y ∨z)∧(¬x∨y ∨¬z)∧(x∨y ∨¬z). 

Then there are four groups of three nodes each. 

Assignment




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

x 

y 

z 

¬x 

¬y z 

¬x 

y 

¬z 

Assignment 

x 

y 

¬z




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

(2) Connect each node in one group with every node in the 

other groups that is logically compatible with it. 

That is, for every variable x, connect x with everything 

except ¬x in every other clause. 

Do this for each group. 

Assignment




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

x 

y 

z 

¬x 

¬y z 

¬x 

y 

¬z 

Assignment 

x 

y 

¬z




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

x 

y 

z 

¬x 

¬y z 

¬x 

y 

¬z 

Assignment 

x 

y 

¬z




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

x 

y 

z 

¬x 

¬y z 

¬x 

y 

¬z 

Assignment 

x 

y 

¬z




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

x 

y 

z 

¬x 

¬y z 

¬x 

y 

¬z 

Assignment 

x 

y 

¬z




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

x 

y 

z 

¬x 

¬y z 

¬x 

y 

¬z 

Assignment 

x 

y 

¬z




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

x 

y 

z 

¬x 

¬y z 

¬x 

y 

¬z 

Assignment 

x 

y 

¬z




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

x 

y 

z 

¬x 

¬y z 

¬x 

y 

¬z 

Assignment 

x 

y 

¬z




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

x 

y 

z 

¬x 

¬y z 

¬x 

y 

¬z 

Assignment 

x 

y 

¬z




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

x 

y 

z 

¬x 

¬y z 

¬x 

y 

¬z 

Assignment 

x 

y 

¬z




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

x 

y 

z 

¬x 

¬y z 

¬x 

y 

¬z 

Assignment 

x 

y 

¬z




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

x 

y 

z 

¬x 

¬y z 

¬x 

y 

¬z 

Assignment 

x 

y 

¬z




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

Let k be the number of clauses in the expression. 

We now ask, does the graph have a clique of size k? 

The 

Reduction of 

3SAT to 

CLIQUE 

Assignment




Robb T. 

Koether 

Does this graph have a clique of size 4? 

Homework 

Review 



¬x 

¬y z 

The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Assignment 

x 

y 

z 

¬x 

y 

¬z 

x 

y 

¬z




Robb T. 

Koether 

Yes it does, namely {x, ¬y, ¬z}. 

Homework 

Review 



¬x 

¬y z 

The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Assignment 

x 

y 

z 

¬x 

y 

¬z 

x 

y 

¬z




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Assignment 

This clique gives us values for x, y, and z that will 

satisfy the expression. 

Namely, x is true, y is false, and z is false. 

This shows that “yes” to CLIQUE implies “yes” to 3SAT. 

It is also easy to see that “no” to CLIQUE implies “no” 

to 3SAT. 

It is also the case that this reduction can be done in 

polynomial time.




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

Theorem 

If B is NP-complete and B ∈ P, then P = NP. 

The 

Reduction of 

3SAT to 

CLIQUE 

Assignment




Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Theorem 

If B is NP-complete and B ≤ P C for some C ∈ NP, then C 

is NP-complete. 

Proof. 

The relation ≤ P is transitive. 

Assignment

Assignment 



Robb T. 

Koether 

Homework 

Review 



The SAT 

Problem 

Cook’s 

Theorem 

The 

Reduction of 

3SAT to 

CLIQUE 

Homework 

Read Section 7.4, pages 271 - 276. 

Exercises 9, 10, page 295. 

Problems 13, 18, page 295. 

Assignment

NP-Completeness - Hampden-Sydney College

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