We show that Closest Substring, one of the most important problems in the eld of biological sequence analysis, is W[1]-hard when parameterized by the number k of input strings (and remains so, even over a binary alphabet). This problem is therefore unlikely to be solvable in time O(f(k) n c ) for any function f of k and constant c independent of 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, although both problems 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 of similar signicance in computational biology.
['Michael R. Fellows', 'Jens Gramm', 'Rolf Niedermeier']
Parameterized Intractability of Motif Search Problems