Median inverse problem and approximating the number of $k$-median inverses of a permutation

TL;DR

This paper introduces the Median Inverse Problem in metric spaces, focusing on permutations under the breakpoint distance, and provides asymptotic bounds on the number of k-median inverses of a permutation.

Contribution

It defines the Median Inverse Problem for permutations and offers asymptotic upper bounds on the number of k-median inverses, advancing understanding of median problems in permutation groups.

Findings

Provides an upper bound for the cardinality of the k-median inverse set.

Derives an asymptotic upper bound for the probability that a permutation is a breakpoint median.

Analyzes median problems in the symmetric group with the breakpoint distance.

Abstract

We introduce the "Median Inverse Problem" for metric spaces. In particular, having a permutation $\pi$ in the symmetric group $S_n$ (endowed with the breakpoint distance), we study the set of all $k$-subsets $\{x_1,...,x_k\}\subset S_n$ for which $\pi$ is a breakpoint median. The set of all $k$-tuples $(x_1,...,x_k)$ with this property is called the $k$-median inverse of $\pi$. Finding an upper bound for the cardinality of this set, we provide an asymptotic upper bound for the probability that $\pi$ is a breakpoint median of $k$ permutations $\xi_1^{(n)},...,\xi_k^{(n)}$ chosen uniformly and independently at random from $S_n$.