Combined super-/substring and super-/subsequence problems

Super-/substring problems and super-/subsequence problems are well-known problems in stringology that have applications in a variety of areas, such as manufacturing systems design and molecular biology. Here we investigate the complexity of a new type of such problem that forms a combination of a super-/substring and a super-/subsequence problem. Moreover we introduce different types of minimal superstring and maximal substring problems. In particular, we consider the following problems: given a set L of strings and a string S, (i) find a minimal superstring (or maximal substring) of L that is also a supersequence (or a subsequence) of S, (ii) find a minimal supersequence (or maximal subsequence) of L that is also a superstring (or a substring) of S. In addition some non-super-/non-substring and non-super-/nonsubsequence variants are studied. We obtain several NP-hardness or even MAX SNP-hardness results and also identify types of "weak minimal" superstrings and "weak maximal" substrings for which (i) is polynomial-time solvable.

[1]  David F. Manlove.  Minimaximal and maximinimal optimisation problems : a partial order-based approach , 1998 .

[2]  Dan Gusfield,et al.  Algorithms on Strings, Trees, and Sequences - Computer Science and Computational Biology , 1997 .

[3]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[4]  Martin Middendorf.  The Shortest Common Nonsubsequence Problem is NP-Complete , 1993, Theor. Comput. Sci..

[5]  Florent Masseglia,et al.  An efficient algorithm for Web usage mining , 1999 .

[6]  Robert W. Irving,et al.  Maximal Common Subsequences and Minimal Common Supersequences , 1994, CPM.

[7]  M. Farach.  Optimal suffix tree construction with large alphabets , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[8]  Martin Farach-Colton,et al.  Optimal Suffix Tree Construction with Large Alphabets , 1997, FOCS.

[9]  Tao Jiang,et al.  Linear approximation of shortest superstrings , 1994, JACM.

[10]  Louxin Zhang.  On the Approximation of Longest Common Nonsupersequences and Shortest Common Nonsubsequences , 1995, Theor. Comput. Sci..

[11]  V. G. Timkovskii.  Complexity of common subsequence and supersequence problems and related problems , 1989 .

[12]  Martin Middendorf.  On Finding Minimal, Maximal, and Consistent Sequences over a Binary Alphabet , 1995, Theor. Comput. Sci..

[13]  Dan Gusfield,et al.  Algorithms on Strings, Trees, and Sequences - Computer Science and Computational Biology , 1997 .

[14]  Tao Jiang,et al.  On the Approximation of Shortest Common Supersequences and Longest Common Subsequences , 1995, SIAM J. Comput..

[15]  David Maier,et al.  On Finding Minimal Length Superstrings , 1980, J. Comput. Syst. Sci..

[16]  Lucas Chi Kwong Hui,et al.  Color Set Size Problem with Application to String Matching , 1992, CPM.

[17]  David Maier,et al.  The Complexity of Some Problems on Subsequences and Supersequences , 1978, JACM.

[18]  Esko Ukkonen,et al.  The Shortest Common Supersequence Problem over Binary Alphabet is NP-Complete , 1981, Theor. Comput. Sci..

[19]  Cameron Bruce Fraser,et al.  Subsequences and Supersequences of Strings , 1995 .

[20]  R. Steele.  Optimization , 2005 .

[21]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .