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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