Some Approximations for Shortest Common Nonsubsequences and Supersequences
SPIRE '08 Proceedings of the 15th International Symposium on String Processing and Information Retrieval
Finding the longest common nonsuperstring in linear time
Information Processing Letters
Improved Approximation Results on the Shortest Common Supersequence Problem
SPIRE '09 Proceedings of the 16th International Symposium on String Processing and Information Retrieval
Improved Algorithms for Finding Consistent Superstrings Based on a New Graph Model
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
On trace inclusion optimization problems
Cybernetics and Systems Analysis
Restricted common superstring and restricted common supersequence
CPM'11 Proceedings of the 22nd annual conference on Combinatorial pattern matching
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For every string inclusion relation there are two optimization problems: find a longest string included in every string of a given finite language, and find a shortest string including every string of a given finite language. As an example, the two well-known pairs of problems, the longest common substring (or subsequence) problem and the shortest common superstring (or supersequence) problem, are interpretations of these two problems. In this paper we consider a class of opposite problems connected with string noninclusion relations: find a shortest string included in no string of a given finite language and find a longest string including no string of a given finite language. The predicate "string $\alpha$ is not included in string $\beta$" is interpreted as either "$\alpha$ is not a substring of $\beta$" or "$\alpha$ is not a subsequence of $\beta$". The main purpose is to determine the complexity status of the string noninclusion optimization problems. Using graph approaches we present polynomial-time algorithms for the first interpretation and NP-hardness proofs for the second. We also discuss restricted versions of the problems, correlations between the string inclusion and noninclusion problems, and generalized problems which are the string inclusion problems for one language and the string noninclusion problems for another.In applications the string inclusion problems are used to find a similarity between any structures which can be represented by strings. Respectively, the noninclusion problems can be used to find a nonsimilarity. Such problems occur in computational molecular biology, data compression, pattern recognition, and flexible manufacturing. The above generalized problems arise naturally in all of these applied areas. Apart from this practical reason, we hope that studying the string noninclusion problems will yield deeper understanding of the string inclusion problems.