I think the best exact solutions for the longest common subsequence problem are the dynamic programming ones. But this is only reasonable for 2-4 strings (0(n^2) to O(n^4)). So people use heuristics. A trivial one is to calculate a simple (tree) clustering with something like UPGMA. For example:
Then you calculate the subsequences of the leafs (a,b) and (c,d) and in the next step (according some special rules) the subsequences of the subsequences.