1. NP-Complete Problems

2. NP and P
What is NP?
NP is the set of all decision problems (question with yes-or-no answer) for which the 'yes'-answers can be verified in polynomial time (O(n^k) where n is the problem size, and k is a constant) by a deterministic Turing machine. Polynomial time is sometimes used as the definition of fast or quickly.
What is P?
P is the set of all decision problems which can be solved in polynomial time by a deterministic Turing machine. Since it can solve in polynomial time, it can also be verified in polynomial time. Therefore P is a subset of NP.

3. Class of “P” Problems
Class P consists of (decision) problems that are solvable in polynomial time
Polynomial-time algorithms
Worst-case running time is O(nk), for some constant k
Examples of polynomial time:
O(n2), O(n3), O(1), O(n lg n)
Examples of non-polynomial time:
O(2n), O(nn), O(n!)

4. Class of “NP” Problems
Class NP consists of problems that could be solved by NP algorithms
i.e., verifiable in polynomial time
If we were given a “certificate” of a solution, we could verify that the certificate is correct in time polynomial to the size of the input
Warning: NP does not mean “non-polynomial”

5. NP-Complete What is NP-Complete?
A problem x that is in NP is also in NP-Complete if and only if every other problem in NP can be quickly (ie. in polynomial time) transformed into x. In other words:
x is in NP, and
Every problem in NP is reducible to x
So what makes NP-Complete so interesting is that if any one of the NP-Complete problems was to be solved quickly then all NP problems can be solved quickly

6. NP-Hard What is NP-Hard?
NP-Hard are problems that are at least as hard as the hardest problems in NP. Note that NP-Complete problems are also NP-hard. However not all NP-hard problems are NP (or even a decision problem), despite having 'NP' as a prefix. That is the NP in NP-hard does not mean 'non-deterministic polynomial time'. Yes this is confusing but its usage is entrenched and unlikely to change.

7. Tractable/Intractable Problems
Problems in P are also called tractable
Problems not in P are intractable or unsolvable
Can be solved in reasonable time only for small inputs
Or, can not be solved at all
Are non-polynomial algorithms always worst than polynomial algorithms?
- n1,000,000 is technically tractable, but really impossible - nlog log log n is technically intractable, but easy

8. Reductions
Reduction is a way of saying that one problem is “easier” than another.
We say that problem A is easier than problem B, (i.e., we write “A  B”)
if we can solve A using the algorithm that solves B.
Idea: transform the inputs of A to inputs of B f
Problem B  
yes no
Problem A

9. Polynomial Reductions
Given two problems A, B, we say that A is polynomially reducible to B (A p B) if:
There exists a function f that converts the input of A to inputs of B in polynomial time
A(i) = YES  B(f(i)) = YES

10. NP-Complete Problems

11. Traveling Salesman

12. 5-Clique

13. Hamiltonian Path

14. Map Coloring

15. Vertex Cover (VC)
Given a graph and an integer k, is there a collection of k vertices such that each edge is connected to one of the vertices in the collection?