The Longest Common Subsequence (or Substring) problem (LCS) asks, given two strings $s_1$ and $s_2$, to find one string which is substring both of $s_1$ and $s_2$ and it is of maximum length.
Each instance of the problem (i.e. each choice of $s_1$ and $s_2$) can have multiple optimal solutions. The size input of the problem is the sum of the length of $s_1$ and $s_2$.
Is the number of optimal solutions polynomial with respect to the size of the instance?