On the subgroup distance problem

We investigate the computational complexity of finding an element of a permutation group H ⊆ S n with minimal distance to a given π ∈ S n , for different metrics on S n . We assume that H is given by a set of generators. In particular, the size of H might be exponential in the input size, so that in general 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 [R.G.E. Pinch, The distance of a permutation from a subgroup of S n , in: G. Brightwell, I. Leader, A. Scott, A. Thomason (Eds.), Combinatorics and Probability, Cambridge University Press, 2007, pp. 473–479]. We present a much simpler proof for this result, which also works for the Hamming Distance, the l p 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 discuss the complexity of the corresponding fixed-parameter and maximization problems.