Sorting by Transpositions Is Difficult
In comparative genomics, a transposition is an operation that exchanges two consecutive sequences of genes in a genome. The transposition distance, that is, the minimum number of transpositions needed to transform a genome into another, can be considered as a relevant evolutionary distance. The problem of computing this distance when genomes are represented by permutations, called the Sorting by Transpositions problem (SBT), has been introduced by Bafna and Pevzner  in 1995. It has naturally been the focus of a number of studies, but the computational complexity of this problem has remained undetermined for 15 years.
In this paper, we answer this long-standing open question by proving that the Sorting by Transpositions problem is NP-hard. As a corollary of our result, we also prove that the following problem from  is NP-hard: given a permutation π, is it possible to sort π using d b (π)/3 permutations, where d b (π) is the number of breakpoints of π?
KeywordsBasic Block Boolean Variable Conjunctive Normal Form Boolean Formula Word Representation
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