# Exam 2 Review

## Reductions

We say that a problem $A$ reduces to a problem $B$ if there is a **polynomial time** reduction function $f$ such that for all $x$, $x \in A \iff f(x) \in B$.

To prove a reduction, we need to show that the reduction function $f$:

1. runs in polynomial time
2. $x \in A \iff f(x) \in B$.

### Useful results from reductions

1. $B$ is at least as hard as $A$ if $A \leq B$.
2. If we can solve $B$ in polynomial time, then we can solve $A$ in polynomial time.
3. If we want to solve problem $A$, and we already know an efficient algorithm for $B$, then we can use the reduction $A \leq B$ to solve $A$ efficiently.
4. If we want to show that $B$ is NP-hard, we can do this by showing that $A \leq B$ for some known NP-hard problem $A$.

$P$ is the class of problems that can be solved in polynomial time. $NP$ is the class of problems that can be verified in polynomial time.

We know that $P \subseteq NP$.

### NP-complete problems

A problem is NP-complete if it is in $NP$ and it is also NP-hard.

#### NP

A problem is in $NP$ if 

1. there is a polynomial size certificate for the problem, and
2. there is a polynomial time verifier for the problem that takes the certificate and checks whether it is a valid solution.

#### NP-hard

A problem is NP-hard if every instance of $NP$ hard problem can be reduced to it in polynomial time.

List of known NP-hard problems:

1. 3-SAT (or SAT)
2. Independent Set
3. Vertex Cover
4. 3-coloring
5. Hamiltonian Cycle
6. Hamiltonian Path

## Approximation Algorithms

## Randomized Algorithms

## Online Algorithms