String Noninclusion Optimization Problems

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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 α is not included in string β” is interpreted as either “α is not a substring of β” or “α is not a subsequence of β”. The main purpose is to determine the complexity status of the string noninclusion optimization problems. Using graph approaches we present polynomi...

Anatoly R. Rubinov, Vadim G. Timkovsky

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