Maximal common subsequences and minimal common supersequences

Abstract

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 number k of strings. Here we study the related problems of finding a minimum-length maximal common subsequence and a maximum-length 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 the minimum maximal common subsequence problem is NP-hard in general for k strings, and we prove a strong negative approximability result for this problem. The complexity of the maximum minimal common supersequence problem for general k remains open, though we conjecture that it too is NP-hard.Key wordsstring algorithmssubsequencesupersequencedynamic programmingNP-hard optimisation problemsapproximation algorithms