The Parameterized Complexity of some Permutation Group Problems · arXivDesk
1301.0379Jan 3, 2013In the revised version the FPT algorithm for computing a base of size k is only for cyclic permutation groups and not for all abelian permutation groups
The Parameterized Complexity of some Permutation Group Problems
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 Sn, and a parameter k, find a permutation π in G such that ∣i∈[n]∣π(i)=i∣
Nearby in the stack
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.