# MAX-SAT 2-Approximation algorithm

I have the two following questions:

1. I know SAT -> MAX-SAT but how can I show that if MAX-SAT is solved in polynomial time then SAT is solved in polynomial time as well?(I guess using approximation algorithms)

2. I have this simple randomized algorithm for approximating the MAX-SAT problem:

The algorithm sets each variable to be True independently with probability 1/2 and False with probability 1/2. It then outputs the resulting assignment.

for(i=1,2,...,n){
set xi = 0 or xi = 1 with probability 1/2 for each case
}


So everywhere I have looked I found people presenting this as an 1/2 approximation algorithm but I'm supposed to prove that this is a 2 approximation algorithm. Can anyone help clarify this?

A $$1/2$$-approximation algorithm and a $$2$$-approximation algorithm are exactly the same thing.
Here are two definitions of a $$\rho$$-approximation algorithm for a maximization problem:
1. On an instance whose optimal value is $$O$$, the algorithm is guaranteed to output a solution whose values is at least $$\rho \cdot O$$.
2. On an instance whose optimal value is $$O$$, the algorithm is guaranteed to output a solution whose values is at least $$O/\rho$$.
The first definition only makes sense when $$\rho \leq 1$$, while the second only makes sense when $$\rho \geq 1$$. Some people prefer the first one, others prefer the second one. Fortunately, you can tell from the value of $$\rho$$ which one is meant (when $$\rho = 1$$, both concepts are the same).