# The simultaneous conjugacy problem in the symmetric group

## TL;DR

This paper presents an efficient algorithm for solving the transitive simultaneous conjugacy problem in the symmetric group, improving upon previous methods and demonstrating promising experimental performance.

## Contribution

The paper introduces a new algorithm that solves the problem in near-optimal time complexity, surpassing earlier approaches and providing empirical evidence of its efficiency.

## Key findings

- Algorithm runs in near-quadratic time with respect to n
- Expected running time is nearly linear in n for fixed d
- New method outperforms previous algorithms on random instances

## Abstract

The transitive simultaneous conjugacy problem asks whether there exists a permutation $\tau \in S_n$ such that $b_j = \tau^{-1} a_j \tau$ holds for all $j = 1,2, \ldots, d$, where $a_1, a_2, \ldots, a_d$ and $b_1, b_2, \ldots, b_d$ are given sequences of $d$ permutations in $S_n$, each of which generates a transitive subgroup of $S_n$. As from mid 70' it has been known that the problem can be solved in $O(dn^2)$ time. An algorithm with running time $O(dn \log(dn))$, proposed in late 80', does not work correctly on all input data. In this paper we solve the transitive simultaneous conjugacy problem in $O(n^2 \log d / \log n + dn\log n)$ time and $O(n^{3/ 2} + dn)$ space. Experimental evaluation on random instances shows that the expected running time of our algorithm is considerably better, perhaps even nearly linear in $n$ at given $d$.