String Noninclusion Optimization Problems

On trace inclusion optimization problems

“…These problems for sets of strings were considered by many authors [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. For the case of a fixed string, Baeza-Yates [4] has introduced the Directed Acyclic Subsequence Graph (DASG), i.e., the minimal partial automaton that is determined by this string and accepts the language of subsequences of this string.…”

“…Using the DASG automaton [8][9][10][11], many problems of matching sets of strings were solved. The mentioned problems in their general statements are NP-hard even for strings [12,18,19].…”

“…As is shown in [18], the problem on the determination of the longest common nonsupersequence for a set of strings L in the alphabet A has a solution if and only if, for each letter a A Î , a string a L a n Î for some n a > 1. It is easy to see that this condition is also necessary and sufficient for the existence of the longest common nonsupertrace.…”

Finding the longest common nonsuperstring in linear time

“…Examples of well-known inclusion related problems that have been studied vigorously to date [2,3,6] are: the longest common substring problem, the longest common subsequence problem, the shortest common superstring problem, and the shortest common supersequence problem. Meanwhile, examples of non-inclusion related problems [4,9] are: the shortest common nonsubsequence problem, the longest common nonsupersequence problem, the shortest common nonsubstring problem, and the longest common nonsuperstring problem.…”

“…Let S denote the set of all the proper suffixes of strings in F . Rubinov and Timkovsky [9] defined a directed graph Γ S = (V , E) for S whose path corresponds to a CNSS of F . When there is a cycle in Γ S , the LCNSS of F does not exist.…”