The quickest algorithm for solving subset sum currently is $2^{n/2}$ (via Wiki). Why doesn't this violate the Exponential Time Hypothesis which states that “there is no family of algorithms that can solve 3-SAT in $2^{o(n)}$ time.”
Couldn't a 3-SAT problem be translated to a subset sum problem in polynomial time and then solved in $2^{n/2}$ time. What am I missing here?
Given a 3SAT problem of size $n$, you can convert that to a subset sum problem of size $n^2$ (roughly). Now you can apply the algorithm for subset sum to solve the subset sum in time $2^{n^2/2}$ (roughly). Remember, the running time depends on the size (length) of the input, and here the size of the input will be $n^2$, yielding the $2^{n^2/2}$ figure.
That will give you an algorithm for solving a 3SAT problem of size $n$ in $2^{n^2/2}$ time (roughly). But that's much worse than the naive algorithm, which takes $2^n$ time: $2^n$ is much smaller than $2^{n^2/2}$. So this reduction does not contradict the exponential time hypothesis: it doesn't give you a way to solve 3SAT faster than $2^n$ time.
