Maximal Common Subsequences and Minimal Common Supersequences

The problems of finding a longest common subsequence and a shortest common supersequence of a set of strings are well known. They can be solved in polynomial time for two strings (in fact the problems are dual in this case), or for any fixed number of strings, by dynamic programming. But both problems are NP-hard in general for an arbitrary numberkof strings. Here we study the related problems of finding a shortest maximal common subsequence and a longest minimal common supersequence. We describe dynamic programming algorithms for the case of two strings (for which case the problems are no longer dual), which can be extended to any fixed number of strings. We also show that both problems are NP-hard in general forkstrings, although the latter problem, unlike shortest common supersequence, is solvable in polynomial time for strings of length 2. Finally, we prove a strong negative approximability result for the shortest maximal common subsequence problem.

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