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The Closest Substring Problem is to decide, for given strings s1, , sk of length at most and numbers m and d, whether there is a length-m string s and length-m substrings si of si, such that s has a Hamming distance of at most d from each si. If we instead require the sum of all the Hamming distances between s and each si to be bounded by d, then it is called the Consensus Patterns Problem. We contribute to the parameterised complexity analysis of these classical NP-hard string problems by investigating the parameter ( m), i.e., the length difference between input and solution strings. For most combinations of ( m) and one of the classical parameters (m, , k, or d), we obtain fixed-parameter tractability. However, even for constant ( m) and constant alphabet size, both problems remain NP-hard. While this follows from known results with respect to the Closest Substring, we need a new reduction in the case of the Consensus Patterns. As a by-product of this reduction, we obtain an exact exponential-time algorithm for both problems, which is based on an alphabet reduction.
ACM Transactions on Computation Theory (TOCT) – Association for Computing Machinery
Published: Sep 18, 2017
Keywords: Parameterised complexity
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