Inapproximability results for the weight problems of subgroup permutation codes

Min-Zheng Shieh*, Shi-Chun Tsai

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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-εn-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.

Original languageEnglish
Article number6249761
Pages (from-to)6907-6915
Number of pages9
JournalIEEE Transactions on Information Theory
Volume58
Issue number11
DOIs
StatePublished - 1 Nov 2012

Keywords

  • Approximation algorithms
  • coding theory
  • computational complexity
  • subgroup code

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