On the Complexity of Some Variations of Sorting by Transpositions

TL;DR

This paper proves that certain genome rearrangement problems involving transpositions and other rearrangements are NP-hard, especially when considering specific weight ratios and fragmentation costs, advancing understanding in computational biology.

Contribution

It establishes NP-hardness for models combining transpositions with reversals, transreversals, and revrevs under certain weight conditions and fragmentation cost functions.

Findings

NP-hardness proven for models with transpositions and reversals

NP-hardness shown for fragmentation-based cost functions

Results apply to weighted rearrangement problems in genomics

Abstract

One of the main challenges in Computational Biology is to find the evolutionary distance between two organisms. In the field of comparative genomics, one way to estimate such distance is to find a minimum cost sequence of rearrangements (large scale mutations) needed to transform one genome into another, which is called the rearrangement distance. In the past decades, these problems were studied considering many types of rearrangements (such as reversals, transpositions, transreversals, and revrevs) and considering the same weight for all rearrangements, or different weights depending on the types of rearrangements. The complexity of the problems involving reversals, transpositions, and both rearrangements is known, even though the hardness proof for the problem combining reversals and transpositions was recently given. In this paper, we enhance the knowledge for these problems by proving that models involving transpositions alongside reversals, transreversals, and revrevs are NP-hard, considering weights w1 for reversals and w2 for the other rearrangements such that w2/w1 is less than or equal to 1.5. In addition, we address a cost function related to the number of fragmentations caused by a rearrangement, proving that the problem of finding a minimum cost sorting sequence, considering the fragmentation cost function with some restrictions, is NP-hard for transpositions and the combination of reversals and transpositions.