A subquadratic algorithm for the simultaneous conjugacy problem

The d $d$-Simultaneous Conjugacy problem in the symmetric group S n ${S}_{n}$ asks whether there exists a permutation tau is an element of S n $\tau \in {S}_{n}$ such that b j = tau - 1 a j tau ${b}_{j}={\tau }<^>{-1}{a}_{j}\tau $ holds for all j = 1 , 2 , horizontal ellipsis , d $j=1,2,\ldots ,d$, where a 1 , a 2 , horizontal ellipsis , a d ${a}_{1},{a}_{2},\ldots ,{a}_{d}$ and b 1 , b 2 , horizontal ellipsis , b d ${b}_{1},{b}_{2},\ldots ,{b}_{d}$ are given sequences of permutations in S n ${S}_{n}$. The time complexity of existing algorithms for solving the problem is O ( d n 2 ) $O(d{n}<^>{2})$. We show that for a given positive integer d $d$ the d $d$-Simultaneous Conjugacy problem in S n ${S}_{n}$ can be solved in o ( n 2 ) $o({n}<^>{2})$ time. Our algorithm solves a number of problems from various fields of mathematics and computer science.