On the Toric Graph as a Tool to Handle the Problem of Sorting by Transpositions

Abstract

To this date, neither a polynomial algorithm to sort a permutation by transpositions has been found, nor a proof that it is an NP-hard problem has been given. Therefore, determining the exact transposition distance d _t (π) of a generic permutation π, relative to the identity, is generally done by an exhaustive search on the space S _n of all permutations of n elements. In a 2001 paper, Eriksson et al. [1] made a breakthrough by proposing a structure named by them as toric graph, which allowed the reduction of the search space, speeding-up the process, such that greater instances could be solved. Surprisingly, Eriksson et al. were able to exhibit a counterexample to a conjecture by Meidanis et al. [2] that the transposition diameter would be equal to the distance of the reverse permutation $\lfloor{n/2}\rfloor+1$. The goal of the present paper is to further study the toric graph, focusing on the case when n + 1 is prime. We observe that the transposition diameter problem for n = 16 is still open. We determine that there are exactly $\frac{n!-n}{n+1} + n$ vertices in the toric graph and find a lower bound $d_t(\pi) \ge \lfloor{n/2}\rfloor$ on the transposition distance for every permutation π in a unitary toric class that is not the identity permutation. We provide experimental data on the exact distance of those permutations to back our conjecture that $d_t(\pi) \le \lfloor{n/2}\rfloor + 1$, where π belongs to a unitary toric class, and that $\lfloor{n/2}\rfloor + 1$ is equal to the transposition diameter when n + 1 is prime.