Suppose we have a hitting set instance $(U, F)$ where $U$ is the universal set and $F$ is the family of subsets of $U$. $F$ has $k$ many sets $=\{f_1, f_2,...,f_k\}$ and $U$ has n many elements. Using DP can we solve this problem in time $O(2^k)$ ignoring the size of $U$?. First we can reduce the number of elements from $n$ to $2^k$, since if we have more than $2^k$ elements, at least two will belong to the same sets of $F$. I don't know this approach is right or wrong, if it is right then how can we apply DP to solve the rest of the problem in time $O(2^k)$, if not kindly give some ideas to approach this problem. If we try bruteforce after reduction, runtime should be $2^{2^k}$.
