On the Subgroup Distance Problem in Cyclic Permutation Groups
Andreas Rosowski
TL;DR
The paper proves that the Subgroup Distance Problem is NP-complete for a range of permutation metrics when the input group is cyclic, covering $H$, Cayley, $l_ obreak t ty$, $l_p$, Lee, Kendall's tau, and Ulam distances. It employs log-space reductions from SAT variants, notably 3-SAT, X3HS, and Not-All-Equal 3SAT, to construct cyclic input groups and gadgets that encode distance constraints. A notable positive result shows that for fixed $k$, the $l_ obreak t ty$-distance problem is solvable in NL for a single generator and NP-complete in the two-generator abelian case, with NL-hardness established via a 2-SAT reduction. Overall, the work delineates a clear boundary between tractable and intractable cases in the subgroup distance setting, highlighting the persistence of hardness even under highly structured group inputs and across a broad spectrum of metrics.
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 $l_\infty(β, α_1^{z_1}α_2^{z_2}) \leq 1$ for permutation $α_1,α_2,β\in S_n$ where $α_1$ and $α_2$ 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 $l_\infty(β, α^z) \leq 1$ for permutations $α,β\in S_n$. 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 $α,β\in S_t$ with $l_\infty(β, α^{t_1}) \leq 1$ and $l_\infty(β, α^{t_2}) \leq 1$.