Parameterized complexity analysis for the Closest String with Wildcards problem

The Closest String problem asks to find a string s which is not too far from each string in a set of m input strings, where the distance is taken as the Hamming distance. This well-studied problem has various applications in computational biology and drug design. In this paper, we introduce a new variant of Closest String where the input strings can contain wildcards that can match any letter in the alphabet, and the goal is to find a solution string without wildcards. We call this problem the Closest String with Wildcards problem, and we analyze it in the framework of parameterized complexity. Our study determines for each natural parameterization whether this parameterization yields a fixed-parameter algorithm, or whether such an algorithm is highly unlikely to exist.