ECCC-Report TR06-146https://eccc.weizmann.ac.il/report/2006/146Comments and Revisions published for TR06-146en-usFri, 01 Dec 2006 22:13:09 +0200
Paper TR06-146
| On the Subgroup Distance Problem |
Christoph Buchheim,
Peter J Cameron,
Taoyang Wu
https://eccc.weizmann.ac.il/report/2006/146We 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.
Fri, 01 Dec 2006 22:13:09 +0200https://eccc.weizmann.ac.il/report/2006/146