Flipping Letters to minimize the Support of a String.

Given a string s on an alphabet Sigma, a word-length k and a budget D, we want to determine the smallest number of distinct k-mers that can be left in s, if we are allowed to replace up to D letters of s. This problem has several parameters, and we discuss its complexity under all sorts of restrictions on the parameters values. We prove that some versions of the problem axe polynomial, while others are NP-hard. We also introduce some Integer Programming formulations to model the NP-hard cases.