, Volume 79, Issue 3, pp 798–813 | Cite as

Parameterized Complexity of Superstring Problems

  • Ivan Bliznets
  • Fedor V. Fomin
  • Petr A. Golovach
  • Nikolay Karpov
  • Alexander S. Kulikov
  • Saket Saurabh


In the Shortest Superstring problem we are given a set of strings \(S=\{s_1, \ldots , s_n\}\) and 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 \(2^{\mathcal {O}(k)} {\text {poly}}(n)\) finds a superstring of length at most \(\ell \) containing at least k strings of S. We complement this by a 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.


Shortest superstring Parameterized complexity Kernelization 


  1. 1.
  2. 2.
    Bellman, R.: Dynamic programming treatment of the travelling salesman problem. J. ACM (JACM) 9(1), 61–63 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Kernelization lower bounds by cross-composition. SIAM J. Discret. Math. 28(1), 277–305 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bulteau, L., Hüffner, F., Komusiewicz, C., Niedermeier, R.: Multivariate algorithmics for NP-hard string problems. Bull. EATCS 114 (2014)Google Scholar
  5. 5.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M.: Parameterized Algorithms. Springer, Berlin (2015)CrossRefzbMATHGoogle Scholar
  6. 6.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, Texts in Computer Science (2013)CrossRefzbMATHGoogle Scholar
  7. 7.
    Edmonds, J.: Maximum matching and a polyhedron with \(0,1\)-vertices. J. Res. Nat. Bur. Stand. Sect. B 69B, 125–130 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Evans, P.A., Wareham, T.: Efficient restricted-case algorithms for problems in computational biology. Algorithms in Computational Molecular Biology: Techniques. Approaches and Applications, pp. 27–49. Wiley, Wiley Series in Bioinformatics (2011)Google Scholar
  9. 9.
    Fomin, F.V., Lokshtanov, D., Panolan, F., Saurabh, S.: Representative sets of product families. In: Algorithms-ESA 2014, pp. 443–454. Springer (2014)Google Scholar
  10. 10.
    Gallant, J., Maier, D., Storer, J.A.: On finding minimal length superstrings. J. Comput. Syst. Sci. 20(1), 50–58 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)zbMATHGoogle Scholar
  12. 12.
    Golovnev, A., Kulikov, A.S., Mihajlin, I.: Solving 3-superstring in \(3^{n/3}\) time. In: Mathematical Foundations of Computer Science 2013, pp. 480–491. Springer (2013)Google Scholar
  13. 13.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. J. Soc. Ind. Appl. Math. 10(1), 196–210 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Karp, R.M.: Dynamic programming meets the principle of inclusion and exclusion. Op. Res. Lett. 1(2), 49–51 (1982)MathSciNetCrossRefzbMATHGoogle 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.
    Williams, R.: Finding paths of length k in \(O^{*}(2^k)\) time. Inf. Process. Lett. 109(6), 315–318 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

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

Personalised recommendations