On the Subgroup Distance Problem in Cyclic Permutation Groups

Abstract: We show that the Subgroup distance problem regarding the Hamming distance, the Cayley distance, the $l_\infty$ distance, the $l_p$ distance (for all $p \geq 1$), the Lee distance, Kendall's tau distance and Ulam's distance is NP-complete when the input group is cyclic. When we restrict the $l_\infty$ distance to fixed values we show that it is NP-complete to decide whether there are numbers $z_1,z_2 \in \mathbb{N}$ such that $l_\infty(\beta, \alpha_1^{z_1}\alpha_2^{z_2}) \leq 1$ for permutation $\alpha_1,\alpha_2,\beta \in S_n$ where $\alpha_1$ and $\alpha_2$ commute. However on the positive side we can show that it can be decided in NL whether there is a number $z \in \mathbb{N}$ such that $l_\infty(\beta, \alpha^z) \leq 1$ for permutations $\alpha,\beta \in S_n$. For the former we provide a tool, namely for all numbers $t_1,t_2,t \in \mathbb{N}$ where $t$ is required to be odd, $0 \leq t_1 < t_2 < t$ and $t_1 \not\equiv t_2 \bmod q$ for all primes $q \mid t$ we give a constructive proof for the existence of permutations $\alpha,\beta \in S_t$ with $l_\infty(\beta, \alpha^{t_1}) \leq 1$ and $l_\infty(\beta, \alpha^{t_2}) \leq 1$.