1. CSCI 2670 Introduction to Theory of Computing
November 22, 2005

2. Agenda Last week Today The Class P The Class NP More on the class NP
November 22, 2005

3. Solving vs. verifying
What if we can’t solve the problem in O(nk) time?
Given a problem and a potential solution, can we verify the solution is correct?
November 22, 2005

4. Example The vertex cover problem
Given a graph G = <V,E> and a number k in N, does there exist a subset V’ of V such that
|V’| = k
For every (u,v)E, either uV’ or vV’
November 22, 2005

5. Vertex cover
November 22, 2005

6. The vertex cover problem
There is no known polynomial solution to the vertex cover problem
What if we have a potential solution
Can we verify it in O(nk) time?
November 22, 2005

7. Verifier M = “On input <V,E,k,V’>
If |V’| ≠ k, reject
For each vertex v in V’
For each edge (u,w) in E
If u = v or w=v, mark (u,w)
If (u,w) is not marked, reject
Accept”
M accepts <V,E,k,V’> if and only if |V’| = k and every edge in E has at least one endpoint in V’
Computational complexity?
O(|V’|×|E|)
November 22, 2005

8. A={w|V accepts <w,c> for some string c}
The class NP
Definition: A verifier for a language A is an algorithm V, where
A={w|V accepts <w,c> for some string c}
The string c is called a certificate of membership in A.
Definition: NP is the class of languages that have polynomial-time verifiers.
These problems have polynomial-time solutions on nondeterministic TM’s.
November 22, 2005

9. Is NP closed under complementation?
For example, can we verify in polynomial time that a graph cannot be 3-colored?
Not obviously
It seems we need to check many 3-colorings before we can conclude that none exist
The 3-coloring problem is in coNP
November 22, 2005

10. What we know NP P
coNP
November 22, 2005

11. What we don’t know Are there any problems here? NP P coNP
November 22, 2005

12. Solving NP problems
The best-known methods for solving problems in NP that are not known to be in P take exponential time
Brute force search
We don’t know if NP is actually in a smaller complexity class
November 22, 2005