Parameterized Complexity of Superstring Problems

  • Ivan Bliznets
  • Fedor V. Fomin
  • Petr A. Golovach
  • Nikolay Karpov
  • Alexander S. Kulikov
  • Saket Saurabh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9133)

Abstract

In the Shortest Superstring problem we are given a set of strings \(S=\{s_1, \ldots , s_n\}\) and an integer \(\ell \) and the question is to decide whether there is a superstring \(s\) of length at most \(\ell \) containing all strings of \(S\) as substrings. We obtain several parameterized algorithms and complexity results for this problem.

In particular, we give an algorithm which in time \(2^{O(k)} {\text {poly}}(n)\) finds a superstring of length at most \(\ell \) containing at least \(k\) strings of \(S\). We complement this by the lower bound showing that such a parameterization does not admit a polynomial kernel up to some complexity assumption. We also obtain several results about “below guaranteed values” parameterization of the problem. We show that parameterization by compression admits a polynomial kernel while parameterization “below matching” is hard.

References

  1. 1.
  2. 2.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bellman, R.: Dynamic programming treatment of the travelling salesman problem. J. ACM (JACM) 9(1), 61–63 (1962)MATHCrossRefGoogle Scholar
  4. 4.
    Bulteau, L., Hüffner, F., Komusiewicz, C., Niedermeier, R.: Multivariate algorithmics for NP-hard string problems. Bull. EATCS 114, 31–73 (2014)Google Scholar
  5. 5.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, Berlin (2013)MATHCrossRefGoogle Scholar
  6. 6.
    Edmonds, J.: Maximum matching and a polyhedron with \(0,1\)-vertices. J. Res. Nat. Bur. Standards Sect. B 69B, 125–130 (1965)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Evans, P.A., Wareham, T.: Efficient restricted-case algorithms for problems in computational biology. In: Zomaya, A.Y., Elloumi, M. (eds.) Algorithms in Computational Molecular Biology: Techniques, Approaches and Applications. Wiley Series in Bioinformatics, pp. 27–49. Wiley, Chichester (2011)Google Scholar
  8. 8.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer-Verlag, Berlin (2006)Google Scholar
  9. 9.
    Gallant, J., Maier, D., Storer, J.A.: On finding minimal length superstrings. J. Comput. Syst. Sci. 20(1), 50–58 (1980)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)MATHGoogle Scholar
  11. 11.
    Golovnev, A., Kulikov, A.S., Mihajlin, I.: Solving SCS for bounded length strings in fewer than \(2^n\) steps. Inf. Process. Lett. 114(8), 421–425 (2014)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Golovnev, A., Kulikov, A.S., Mihajlin, I.: Solving 3-superstring in 3\(^\text{ n/3 }\) time. In: Chatterjee, K., Sgall, J. (eds.) MFCS 2013. LNCS, vol. 8087, pp. 480–491. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  13. 13.
    Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. J. Soc. Ind. Applied Math. 10(1), 196–210 (1962)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Karp, R.M.: Dynamic programming meets the principle of inclusion and exclusion. Oper. Res. Lett 1(2), 49–51 (1982)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Knuth, D.E., Morris, J.H.J., Pratt, V.R.: Fast pattern matching in strings. SIAM J. Comput. 6(2), 323–350 (1977)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Kohn, S., Gottlieb, A., Kohn, M.: A generating function approach to the traveling salesman problem. In: Proceedings of the 1977 Annual Conference, pp. 294–300. ACM (1977)Google Scholar
  17. 17.
    Mucha, M.: Lyndon words and short superstrings. In: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 958–972. SIAM (2013)Google Scholar
  18. 18.
    Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and near-optimal derandomization. In: FOCS, pp. 182–191. IEEE Computer Society (1995)Google Scholar
  19. 19.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications, vol. 31. Oxford University Press, Oxford (2006) MATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ivan Bliznets
    • 2
  • Fedor V. Fomin
    • 1
    • 2
  • Petr A. Golovach
    • 1
    • 2
  • Nikolay Karpov
    • 2
  • Alexander S. Kulikov
    • 2
  • Saket Saurabh
    • 1
    • 3
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.Steklov Institute of Mathematics at St. PetersburgRussian Academy of SciencesSt. PetersburgRussia
  3. 3.Institute of Mathematical SciencesChennaiIndia

Personalised recommendations