NP-Completeness: A question about reduction and hardness I am trying to understand the definition / meaning of reduction.


Is it correct to say that the statement "Problem $A$ reduces to Problem $B$ in $x$-time" is the same as writing $A \leq_{x} B$? For example $\text{SAT} \leq_{polynomial} \text{3-SAT}$.
If we are reducing problem $A$ to $B$ in $x$-time, does it mean that you take an instance in problem $A$, and modify it such that it becomes a valid input for problem $B$, such that the modification is done with $x$ time complexity?
As far as I know, the first point is correct. For the second point you missed the fact that "solve $A$ with the given instance iff solve $B$ with the created instance".