Inapproximability Results for the Weight Problems of Subgroup Permutation Codes

A subgroup permutation code is a set of permutations on
n
symbols with the property that its elements are closed under the operation of composition. In this paper, we give inapproximability results for the minimum and maximum weight problems of subgroup permutation codes under several well-known metrics. Based on previous works, we prove that under Hamming, Lee, Cayley, Kendall's tau, Ulam's, and
lp
distance metrics, 1) there is no polynomial-time 2
^log1-ε
n
-approximation algorithm for the minimum weight problem for any constant ε >; 0 unless NP ⊆ DTIME(2
^polylog(
n
)) (quasi-polynomial time), and 2) there is no polynomial-time
r
-approximation algorithm for the minimum weight problem for any constant
r
>; 1 unless P = NP. Under
l
_∞
-metric, we prove that it is NP-hard to approximate the minimum weight problem within factor 2-ε for any constant ε >; 0. We also prove that for any constant ε >; 0, it is NP-hard to approximate the maximum weight within
p
√{[ 3/ 2]}-ε under ℓ
_p
distance metric, and within [ 3/ 2]-ε under Hamming, Lee, Cayley, Kendall's tau, and Ulam's distance metrics.