On the parameterized intractability of CLOSEST SUBSTRING and related problems

Abstract

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)nc) for any function f and 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 pa-rameterizations 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.