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

Abstract

The C losest S ubstring P roblem is to decide, for given strings s 1 , … , s k of length at most ℓ and numbers m and d , whether there is a length- m string s and length- m substrings s ′ i of s i , such that s has a Hamming distance of at most d from each s ′ i . If we instead require the sum of all the Hamming distances between s and each s ′ i to be bounded by d , then it is called the C onsensus P atterns P roblem . We contribute to the parameterised complexity analysis of these classical NP-hard string problems by investigating the parameter (ℓ − m ), i.e., the length difference between input and solution strings. For most combinations of (ℓ − m ) and one of the classical parameters ( m , ℓ, k , or d ), we obtain fixed-parameter tractability. However, even for constant (ℓ − m ) and constant alphabet size, both problems remain NP-hard. While this follows from known results with respect to the C losest S ubstring , we need a new reduction in the case of the C onsensus P atterns . As a by-product of this reduction, we obtain an exact exponential-time algorithm for both problems, which is based on an alphabet reduction.