Here's the shortest superstring problem: you are given $n$ strings $s_1,\ldots, s_n$ over some alphabet $\Sigma$ and you want to find the shortest string over $\Sigma$ that contains each $s_i$ as a subsequence of consecutive characters, i.e. a substring.
When we talk about exact algorithms for the problem, finding the length $L$ of the shortest superstring is equivalent to finding the maximum compression $C$ which is the sum of all consecutive string overlaps in the final superstring, i.e. $C=\sum_i |s_i|-L$.
As far as I know, the fastest exact algorithm for shortest superstring runs in $O^*$($2^n$) where $n$ is the number of strings. This is a simple dynamic programming algorithm similar to the dynamic programming algorithm for longest path (and other problems):
For each subset of strings $S$ and string $v$ in $S$ we compute the maximum compression over all superstrings over $S$ where $v$ is the first string appearing in the superstring, storing this as C(($v,S$)). We do this by first processing all subsets with only one element, and then building up the C(($v,S$)) values for subsets $S$ on $k$ strings from those on $k-1$ strings. Specifically:
For each string $u$ we look at all subsets $S'$ on $k-1$ strings that don't include $u$ and set the value for ($u,{u}\cup S'$) to the maximum over
strings $v$ in $S'$ of the sum of the maximum overlap of $u$ with $v$ with C(($v,S'$)).
The final runtime is no more than O($n^2 2^n + n^2 l$) where $l$ is the maximum string length.
There are better algorithms if you assume that $l$ is small, or the pairwise overlaps are small, the alphabet size is small etc, but I am not aware of any algorithm that's faster than $2^n$.