Finding Consensus Strings with Small Length Difference Between Input and Solution Strings

Abstract

The parameterised complexity of the Closest Substring Problem and the Consensus Patterns Problem with respect to the parameter $(\ell - m)$ is investigated, where $\ell$ is the maximum length of the input strings and m is the length of the solution string. We present an exact exponential time algorithm for both problems, which is based on an alphabet reduction. Furthermore, it is shown that for most combinations of $(\ell - m)$ and one of the classical parameters (m, $\ell$, number of input strings k, distance d), we obtain fixed-parameter tractability, but even for constant $(\ell - m)$ and constant alphabet size, both problems are ${{\mathrm{\mathsf {NP}}}}$-hard.