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Minimum Common String Partition Problem: Hardness and Approximations

  • Avraham Goldstein
  • Petr Kolman
  • Jie Zheng
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3341)

Abstract

String comparison is a fundamental problem in computer science, with applications in areas such as computational biology, text processing or compression. In this paper we address the minimum common string partition problem, a string comparison problem with tight connection to the problem of sorting by reversals with duplicates, a key problem in genome rearrangement.

A partition of a string A is a sequence \({\mathcal P}=(P_{1},P_{2},...P_{m})\) of strings, called the blocks, whose concatenation is equal to A. Given a partition \({\mathcal P}\) of a string A and a partition \({\mathcal Q}\) of a string B, we say that the pair \(\langle\mathcal{P,Q}\rangle\) is a common partition of A and B if \({\mathcal Q}\) is a permutation of \({\mathcal P}\). The minimum common string partition problem (MCSP) is to find a common partition of two strings A and B with the minimum number of blocks. The restricted version of MCSP where each letter occurs at most k times in each input string, is denoted by k-MCSP.

In this paper, we show that 2-MCSP (and therefore MCSP) is NP-hard and, moreover, even APX-hard. We describe a 1.1037-approximation for 2-MCSP and a linear time 4-approximation algorithm for 3-MCSP. We are not aware of any better approximations.

Keywords

Genome Rearrangement Truth Assignment Small Instance Input String Minimum Vertex Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Avidor, A., Zwick, U.: Approximating MIN k-SAT. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 465–475. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Berman, P., Karpinski, M.: On some tighter inapproximability results. In: Proc. of the of 26th International Colloquium on Automata, Languages and Programming (ICALP), pp. 200–209 (1999)Google Scholar
  3. 3.
    Caprara, A.: Sorting by reversals is difficult. In: Proc. of the First International Conference on Computational Molecular Biology, pp. 75–83 (1997)Google Scholar
  4. 4.
    Chen, X., Zheng, J., Fu, Z., Nan, P., Zhong, Y., Lonardi, S., Jiang, T.: Assignment of orthologous genes via genome rearrangement. Submitted (2004)Google Scholar
  5. 5.
    Christie, D.A., Irving, R.W.: Sorting strings by reversals and by transpositions. SIAM Journal on Discrete Mathematics 14(2), 193–206 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chrobak, M., Kolman, P., Sgall, J.: The greedy algorithm for the minimum common string partition problem. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 84–95. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Cormode, G., Muthukrishnan, S.: The string edit distance matching problem with moves. In: Proc. of the 13th Annual ACM-SIAM Symposium On Discrete Mathematics (SODA), pp. 667–676 (2002)Google Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman & Company, San Francisco (1978)Google Scholar
  9. 9.
    Hannenhalli, S., Pevzner, P.A.: Transforming cabbage into turnip: polynomial algorithm for sorting signed permutations by reversals. Journal of the ACM 46(1), 1–27 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Sankoff, D., El-Mabrouk, N.: Genome rearrangement. In: Jiang, T., Xu, Y., Zhang, M.Q. (eds.) Current Topics in Computational Molecular Biology, MIT Press, Cambridge (2002)Google Scholar
  11. 11.
    Shapira, D., Storer, J.A.: Edit distance with move operations. In: Apostolico, A., Takeda, M. (eds.) CPM 2002. LNCS, vol. 2373, p. 85. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    Watterson, G.A., Ewens, W.J., Hall, T.E., Morgan, A.: The chromosome inversion problem. Journal of Theoretical Biology 99, 1–7 (1982)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Avraham Goldstein
    • 1
  • Petr Kolman
    • 2
  • Jie Zheng
    • 3
  1. 1.No Institute Given 
  2. 2.Institute for Theoretical Computer ScienceCharles UniversityPraha 1Czech Republic
  3. 3.Department of Computer ScienceUniversity of CaliforniaRiverside

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