On The Parameterized Intractability Of Motif Search Problems*

We show that Closest Substring, one of the most important problems in the field of consensus string analysis, is W[1]-hard when parameterized by the number k of input strings (and remains so, even over a binary alphabet). This is done by giving a “strongly structure-preserving” reduction from the graph problem Clique to Closest Substring. This problem is therefore unlikely to be solvable in time O(f(k)•nc) for any function f of k and constant c independent of k, i.e., the combinatorial explosion seemingly inherent to this NP-hard problem cannot be restricted to parameter k. The problem can therefore 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, althoughb othp roblems are NP-complete. We also prove W[1]-hardness for other parameterizations in the case of unbounded alphabet size. Our W[1]-hardness result for Closest Substring generalizes to Consensus Patterns, a problem arising in computational biology.

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Correspondence to Michael R. Fellows.

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* An extended abstract of this paper was presented at the 19th International Symposium on Theoretical Aspects of Computer Science (STACS 2002), Springer-Verlag, LNCS 2285, pages 262–273, held in Juan-Les-Pins, France, March 14–16, 2002.

† Work was supported by the Deutsche Forschungsgemeinschaft (DFG), research project “OPAL” (optimal solutions for hard problems in computational biology), NI 369/2.

‡ Work was done while the author was with Wilhelm-Schickard-Institut für Informatik, Universität Tübingen. Work was partially supported by the Deutsche Forschungsgemeinschaft (DFG), Emmy Noether research group “PIAF” (fixed-parameter algorithms), NI 369/4.

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Fellows, M.R., Gramm†, J. & Niedermeier‡, R. On The Parameterized Intractability Of Motif Search Problems*. Combinatorica 26, 141–167 (2006). https://doi.org/10.1007/s00493-006-0011-4

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Mathematics Subject Classification (2000):

  • 03D15
  • 68Q17
  • 68Q25