The Parameterized Complexity of Fixpoint Free Elements and Bases in Permutation Groups

In this paper we study the parameterized complexity of two well-known permutation group problems which are NP-complete.

1

Given a permutation group

$G=\angle{S}\le S_n$

and a parameter

k

, find a permutation

π

 ∈ 

G

such that |{

i

 ∈ [

n

]|

π

(

i

) ≠ 

i

}| ≥ 

k

. This generalizes the

NP

-complete problem of finding a fixed-point free permutation in

G

[7,14] (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 [8,7].

2

A

base

for

G

is a subset

B

 ⊆ [

n

] such that the subgroup of

G

that fixes

B

pointwise is trivial. We consider the parameterized complexity of checking if a given permutation group

$G=\angle{S}\le S_n$

has a base of size

k

, where

k

is the parameter for the problem. This problem is known to be

NP

-complete [4]. We show that it is fixed-parameter tractable for 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.