On the Parameterized Intractability of Closest Substring and Related Problems

  • Michael R. Fellows
  • Jens Gramm
  • Rolf Niedermeier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2285)


We show that Closest Substring, one of the most important problems in the field of biological sequence analysis, is W[1]-hard with respect to the number k of input strings (even over a binary alphabet). This problem is therefore unlikely to be solvable in time O(f(k)n c) for any function fand constant c independent of k— effectively, the problem can be expected to be intractable, in any practical sense, for k ≥ 3. Our result supports the intuition that Closest Substring is computationally much harder than the special case of Closest String, although both problems are NP-complete and both possess polynomial time approximation schemes. We also prove W[1]-hardness for other parameterizations in the case of unbounded alphabet size. Our main W[1]- hardness result generalizes to CONSENSUS PATTERNS, a problem of similar significance in computational biology.


Close String Vertex Cover Input Graph Solution String Input String 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Blanchette. Algorithms for phylogenetic footprinting. In Proc. of 5th ACM RECOMB, pages 49–58, 2001, ACM Press.Google Scholar
  2. 2.
    H. L. Bodlaender, R. G. Downey, M. R. Fellows, and H. T. Wareham. The parameterized complexity of sequence alignment and consensus. Theoretical Computer Science, 147:31–54, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    H. L. Bodlaender, R. G. Downey, M. R. Fellows, M. T. Hallett, and H. T. Wareham. Parameterized complexity analysis in computational biology. Computer Applications in the Biosciences, 11: 49–57, 1995.Google Scholar
  4. 4.
    M. Cesati and L. Trevisan. On the efficiency of polynomial time approximation schemes. Information Processing Letters, 64(4):165–171, 1997.CrossRefMathSciNetGoogle Scholar
  5. 5.
    R. G. Downey and M. R. Fellows. ParameterizedComplexity. Springer. 1999.Google Scholar
  6. 6.
    P. A. Evans and H. T. Wareham. Practical non-polynomial time algorithms for designing universal DNAoligon ucleotides: a systematic approach. Manuscript, April 2001.Google Scholar
  7. 7.
    M. Frances and A. Litman. On covering problems of codes. Theory of Computing Systems, 30:113–119, 1997.zbMATHMathSciNetGoogle Scholar
  8. 8.
    J. Gramm, R. Niedermeier, and P. Rossmanith. Exact solutions for Closest String and related problems. To appear in Proc. of 12th ISAAC (Christchurch, New Zealand), LNCS, December 2001. Springer.Google Scholar
  9. 9.
    M. T. Hallett. An IntegratedComplexity Analysis of Problems from Computational Biology. PhD Thesis, University of Victoria, Canada, 1996.Google Scholar
  10. 10.
    J. K. Lanctot, M. Li, B. Ma, S. Wang, and L. Zhang. Distinguishing string selection problems. In Proc. of 10th ACM-SIAM SODA, pages 633–642, 1999, ACM Press. To appear in Information andComputation.Google Scholar
  11. 11.
    M. Li, B. Ma, and L. Wang. Finding similar regions in many strings. In Proc. of 31st ACM STOC, pages 473–482, 1999. ACM Press. To appear in Journal of Computer andSystem Sciences.Google Scholar
  12. 12.
    B. Ma. A polynomial time approximation scheme for the closest substring problem. In Proc. of 11th CPM, number 1848 in LNCS, pages 99–107, 2000. Springer.Google Scholar
  13. 13.
    P. A. Pevzner. Computational Molecular Biology — An Algorithmic Approach. MITPress. 2000.Google Scholar
  14. 14.
    P. A. Pevzner and S.-H. Sze. Combinatorial approaches to finding subtle signals in DNA sequences. In Proc. of 8th ISMB, pages 269–278, 2000. AAAI Press.Google Scholar
  15. 15.
    M.-F. Sagot. Spelling approximate repeated or common motifs using a sufix tree. In Proc. of 3rdLA TIN, number 1380 in LNCS, pages 111–127, 1998. Springer.Google Scholar
  16. 16.
    D. Sanko. and J. Kruskal (eds.). Time Warps, String Edits, and Macromolecules. Addison-Wesley. 1983. Reprinted in 1999 by CSLI Publications.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Michael R. Fellows
    • 1
  • Jens Gramm
    • 2
  • Rolf Niedermeier
    • 2
  1. 1.Department of Computer Science and Software EngineeringUniversity of Newcastle, University DriveCallaghanAustralia
  2. 2.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenFed. Rep. of Germany

Personalised recommendations