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:

$a='AAAB';
$b='AABB';
$c='ABBB';
$d='BBBB';

     +---$a 
+----+
|    +---$b 
|
+
|
|    +---$c 
+----+
     +---$d
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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.