Inapproximability Results for the Weight Problems of Subgroup Permutation Codes

Abstract

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 l(p) distance metrics, 1) there is no polynomial-time 2(log1-epsilon n)-approximation algorithm for the minimum weight problem for any constant epsilon > 0 unless NP subset of 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-infinity-metric, we prove that it is NP-hard to approximate the minimum weight problem within factor 2 - epsilon for any constant epsilon > 0. We also prove that for any constant epsilon > 0, it is NP-hard to approximate the maximum weight within p root 3/2 - epsilon under l(p) distance metric, and within 3/2 - epsilon under Hamming, Lee, Cayley, Kendall's tau, and Ulam's distance metrics. Index Terms-Approximation algorithms,