An algebraic 1.375-approximation algorithm for the Transposition Distance Problem
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In genome rearrangements, the mutational event transposition swaps two adjacent blocks of genes in one chromosome. The Transposition Distance Problem (TDP) aims to find the minimum number of transpositions (distance) required to transform one chromosome into another, both represented as permutations. Setting the target permutation as the identity permutation, makes the TDP equivalent to the problem of Sorting by Transpositions (SBT). TDP is NP-hard and the best approximation algorithm with a 1.375 ratio was proposed in 2006 by Elias and Hartman. Their algorithm employs a technique called simplification to transform an input permutation π into a simple permutationπ̂, obtained by inserting new symbols into π in a way that the lower bound of the transposition distance of π is kept on π̂. The assumption is that handling with simple permutations is easier than with normal ones. A sequence of transpositions sorting π̂ can be mimicked to sort π. In this work, we first show that the algorithm of Elias and Hartman may require one transposition above the intended approximation ratio of 1.375, depending on how the input permutation is simplified. Then, we propose a new 1.375-approximation algorithm to solve TDP based on an algebraic formalism which does not use simplification, ensuring the approximation ratio of 1.375 for all the permutations in the Symmetric Group S_n. We also propose a new upper bound for the transposition distance. Implementations of our algorithm and the one of Elias and Hartman were audited using GRAAu tool. The results show that, taking as input short permutations with maximum length 13, in addition to keeping the approximation below the 1.375 ratio, our algorithm returns a higher percentage of exact distances than the one of Elias and Hartman.
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