Finding cycle types in permutation groups with few generators
Markus Lohrey, Andreas Rosowski
TL;DR
This paper addresses the CycleType problem for permutation groups: given a group $G \leq \mathsf{Sym}(n)$ described by generators and a target cycle type, determine if $G$ contains an element of that type. It shows a sharp boundary: CycleType can be solved in logspace when $G$ is cyclic, but becomes NP-complete already for a 2-generated abelian (commuting) group; similarly, FixpointFree is NP-complete for 2-generated abelian groups and for certain coset variants. The hardness results arise from logspace reductions from X3HS (positive 1-in-3-SAT) that encode cycle-structure with primes and carefully constructed gadget permutations. Together, these results illuminate the tractability frontier for cycle-type problems under very small generator sets in permutation groups, with implications for related derangement problems and coset computations.
Abstract
The problem whether a given permutation group contains a permutation with a given cycle type is studied. This problem is known to be NP-complete. In this paper it is shown that the problem can be solved in logspace for a cyclic permutation group and that it is NP-complete for a 2-generated abelian permutation group. In addition it is shown that it is NP-complete whether a 2-generated abelian permutation group contains a fixpoint-free permutation.