The complexity of the weight problem for permutation and matrix groups

Given a metric d on a permutation group G , the corresponding weight problem is to decide whether there exists an element π ∈ G such that d ( π , e ) = k , for some given value k . Here we show that this problem is NP-complete for many well-known metrics. An analogous problem in matrix groups, eigenvalue-free problem, and two related problems in permutation groups, the maximum and minimum weight problems, are also investigated in this paper.