Abstract:
We investigate the computational complexity of finding an element of a permutation group~$H\subseteq S_n$ with a minimal distance to a given~$\pi\in S_n$, for different metrics on~$S_n$. 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~\cite{pinch06}. 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 $l_\infty$ distance using a different reduction idea. Finally, we settle the complexity of the corresponding fixed-parameter and maximization problems.