Permutation-constrained Common String Partitions with Applications

We study a new combinatorial problem based on the famous Minimum Common String Partition problem, which we call Permutation-constrained Common String Partition (PCSP for short). In PCSP, we are given two sequences/genomes s and t with the same length and a permutation \(\pi \) on \([\ell ]\), the question is to decide whether it is possible to decompose s and t into \(\ell \) blocks that can be matched according to some specified requirements, and that conform with the permutation \(\pi \). Our main result is that PCSP is FPT in parameter \(\ell + d\), where d is the maximum number of occurrences that any symbol may have in s or t. We also study a variant where the input specifies whether each matched pair of block needs to be preserved as is, or reversed. With this result on PCSP, we show that a series of genome rearrangement problems are FPT \(k + d\), where k is the rearrangement distance between two genomes of interest.