A 1.375-Approximation Algorithm for Sorting by Transpositions

Abstract

Abstract. Sorting permutations by transpositions is an important problem in genome rearrangements. A transposition is a rearrangement operation in which a segment is cut out of the permutation and pasted in a different location. The complexity of this problem is still open and it has been a ten-year-old open problem to improve the best known 1.5-approximation algorithm. In this paper we provide a 1.375-approximation algorithm for sorting by transpositions. The algorithm is based on a new upper bound on the diameter of 3-permutations. In addition, we present some new results regarding the transposition diameter: We improve the lower bound for the transposition diameter of the symmetric group, and determine the exact transposition diameter of 2-permutations and simple permutations.

Expected number of breakpoints after t random reversals in genomes with duplicate genes

“…For each set of operations, such as reversals, block transpositions and others, including combinations, one defines the distance d(π, τ ) as the minimal number of operations needed to transform π into τ . There are many papers devoted to computing such distances, for example [13,6,8]. The operations may also be weighted in different ways [5,10].…”

Sorting by Genome Rearrangements on Both Gene Order and Intergenic Sizes

“…When the orientation of genes is unknown, it has been proved that the Sorting Permutations by Reversals problem is NP-hard (Caprara, 1999), and the best approximation algorithm has an approximation factor of 1.375 (Berman et al, 2002). The Sorting Permutations by Transpositions problem is also NP-hard (Bulteau et al, 2012), and the best approximation algorithm has an approximation factor of 1.375 (Elias and Hartman, 2006). If the model M allows both reversals and transpositions, the problem is called Sorting Permutations by Reversals and Transpositions.…”

Ordenação de Permutações com Sinais por Reversões e Transposições

“…Quando consideramos o mesmo modelo, mas utilizando permutações sem sinais, foi provado que o problema pertenceà classe de problemas NP-Difícil [Caprara 1999] e o melhor algoritmo conhecido possui um fator de aproximação 1.375 [Berman et al 2002]. De maneira similar, quando consideramos apenas o evento de transposição e permutações sem sinais, o problema também pertenceà classe de problemas NP-Difícil [Bulteau et al 2012] e o melhor algoritmo conhecido possui um fator de aproximação 1.375 [Elias and Hartman 2006].…”