Narrow Results

We consider the problem BLOCK-SORTING: Given a permutation, sort it by using a minimum number of block moves, where a block is a maximal substring of the permutation which is also a substring of the identity permutation, and a block move repositions the chosen block so that it merges with another block. Although this problem has recently been shown to be NP-hard [3], nothing better than a trivial 3-approximation was known. We present here the first non-trivial approximation algorithm to this problem. For this purpose, we introduce the following optimization problem: Given a set of increasing sequences of distinct elements, merge them into one increasing sequence with a minimum number of block moves. We show that the merging problem has a polynomial time algorithm. Using this, we obtain an O(n³) time 2-approximation algorithm for BLOCK-SORTING.

We also observe that BLOCK-SORTING, as well as sorting by transpositions, are fixed-parameter-tractable in the framework of [6].

Some interesting combinatorial problems have been motivated by genome rearrangements, which are mutations that affect large portions of a genome. When we represent genomes as permutations, the goal is to transform a given permutation into the identity permutation with the minimum number of rearrangements. When they affect segments from the beginning (respectively end) of the permutation, they are called prefix (respectively suffix) rearrangements. This paper presents results for rearrangement problems that involve prefix and suffix versions of reversals and transpositions considering unsigned and signed permutations. We give 2-approximation and ($2+\lambda$)-approximation algorithms for these problems, where $\lambda$ is a constant divided by the number of breakpoints (pairs of consecutive elements that should not be consecutive in the identity permutation) in the input permutation. We also give bounds for the diameters concerning these problems and provide ways of improving the practical results of our algorithms.

One of the main problems in Computational Biology is to find the evolutionary distance among species. In most approaches, such distance only involves rearrangements, which are mutations that alter large pieces of the species’ genome. When we represent genomes as permutations, the problem of transforming one genome into another is equivalent to the problem of Sorting Permutations by Rearrangement Operations. The traditional approach is to consider that any rearrangement has the same probability to happen, and so, the goal is to find a minimum sequence of operations which sorts the permutation. However, studies have shown that some rearrangements are more likely to happen than others, and so a weighted approach is more realistic. In a weighted approach, the goal is to find a sequence which sorts the permutations, such that the cost of that sequence is minimum. This work introduces a new type of cost function, which is related to the amount of fragmentation caused by a rearrangement. We present some results about the lower and upper bounds for the fragmentation-weighted problems and the relation between the unweighted and the fragmentation-weighted approach. Our main results are 2-approximation algorithms for five versions of this problem involving reversals and transpositions. We also give bounds for the diameters concerning these problems and provide an improved approximation factor for simple permutations considering transpositions.

Problems in the genome rearrangement field are often formulated in terms of pairwise genome comparison: given two genomes $G_1$ and $G_2$, find the minimum number of genome rearrangements that may have occurred during the evolutionary process. This broad definition lacks at least two important considerations: the first being which features are extracted from genomes to create a useful mathematical model, and the second being which types of genome rearrangement events should be represented. Regarding the first consideration, seminal works in the genome rearrangement field solely used gene order to represent genomes as permutations of integer numbers, neglecting many important aspects like gene duplication, intergenic regions, and complex interactions between genes. Regarding the second consideration, some rearrangement events are widely studied such as reversals and transpositions. In this paper, we shed light on the first consideration and created a model that takes into account gene order and the number of nucleotides in intergenic regions. In addition, we consider events of reversals, transpositions, and indels (insertions and deletions) of genomic material. We present a 4-approximation algorithm for reversals and indels, a $4.5$-approximation algorithm for transpositions and indels, and a 6-approximation for reversals, transpositions, and indels.

In this paper, we explain why the chaotic mutation (CM) model of J. M. Bahi and C. Michel (2008) simulates the genes mutations over time with good accuracy. It is firstly shown that the CM model is a truly chaotic one, as it is defined by Devaney. Then, it is established that mutations occurring in genes mutations have indeed a same chaotic dynamic, thus making relevant the use of chaotic models for genomes evolution. Transposition and inversion dynamics are finally investigated.