On the Subgroup Distance Problem

We investigate the computational complexity of finding an element of a permutation group H ⊆ Sn with a minimal distance to a given π ∈ Sn, for different metrics on Sn. We assume that H is given by a set of generators, such that the problem cannot be solved in polynomial time by exhaustive enumeration. For the case of the Cayley Distance, this problem has been shown to be NP-hard, even if H is abelian of exponent two [7]. We present a much simpler proof for this result, which also works for the Hamming Distance, the lp distance, Lee’s Distance, Kendall’s tau, and Ulam’s Distance. Moreover, we give an NP-hardness proof for the l ∞ distance using a different reduction idea. Finally, we settle the complexity of the corresponding fixed-parameter and maximization problems. Key words: permutation groups, subgroup distanc