A new 1.375-approximation algorithm for Sorting By Transpositions

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 required to transform one chromosome into another, both represented as permutations. The TDP can be reduced to the problem of Sorting by Transpositions (SBT). SBT is $\mathcal{NP}$-hard and the best approximation algorithm with a $1.375$ ratio was proposed by Elias and Hartman. Their algorithm employs simplification, a technique used to transform an input permutation $\pi$ into a simple permutation $\hat{\pi}$, presumably easier to handle with. The permutation $\hat{\pi}$ is obtained by inserting new symbols into $\pi$ in a way that the lower bound of the transposition distance of $\pi$ is kept on $\hat{\pi}$. The simplification is guaranteed to keep the lower bound, not the transposition distance. In this paper, we first show that the algorithm of Elias and Hartman (EH algorithm) may require one extra transposition above the approximation ratio of $1.375$, depending on how the input permutation is simplified. Next, using an algebraic approach, we propose a new upper bound for the transposition distance and a new $1.375$-approximation algorithm to solve SBT skipping simplification and ensuring the approximation ratio of $1.375$ for all $S_n$. We implemented our algorithm and EH's. Regarding the implementation of the EH algorithm, two issues needed to be fixed. We tested both algorithms against all permutations of size $n$, $2\leq n \leq 12$. The results show that the EH algorithm exceeds the approximation ratio of $1.375$ for permutations with a size greater than $7$. Finally, we investigate the performance of both implementations on longer permutations of maximum length $500$.