The Parameterized Complexity of some Permutation Group Problems

TL;DR

This paper investigates the parameterized complexity of two permutation group problems, providing fixed-parameter tractability results for specific cases and solving an open problem about fixed points.

Contribution

It demonstrates fixed-parameter tractability for finding permutations with many fixed points and for computing bases in certain permutation groups, and resolves an open question on fixed points.

Findings

Fixed-parameter tractability for finding permutations with at least k fixed points.

A simple polynomial-time algorithm for fixed point free elements in transitive groups.

Fixed-parameter tractability for bases in cyclic and constant orbit size groups.

Abstract

In this paper we study the parameterized complexity of two well-known permutation group problems which are NP-complete. 1. Given a permutation group G=<S>, subgroup of $S_n$, and a parameter $k$, find a permutation $\pi$ in G such that $|{i\in [n]\mid \pi(i)\ne i}|$ is at least $k$. This generalizes the well-known NP-complete problem of finding a fixed-point free permutation in G. (this is the case when $k=n$). We show that this problem with parameter $k$ is fixed parameter tractable. In the process, we give a simple deterministic polynomial-time algorithm for finding a fixed point free element in a transitive permutation group, answering an open question of Cameron. 2. Next we consider the problem of computing a base for a permutation group G=<S>. A base for G is a subset B of $[n]$ such that the subgroup of G that fixes B pointwise is trivial. This problem is known to be NP-complete. We show that it is fixed parameter tractable for the case of cyclic permutation groups and for permutation groups of constant orbit size. For more general classes of permutation groups we do not know whether the problem is in FPT or is W[1]-hard.