On the dependence of the maximum cycle mean of a matrix on permutations of the rows and columns

Let (G, -, 5) be a radicable, linearly ordered, commutative group. Given a square matrix A = (a$ of order n with entries from G and a cyclic permutation o = (il. . . . , i,) of a subset of N=I1,2 . . . ,n} we define p(a), the mean weight of o, as (ai,iz * ai2i3 *-a ai,_ ,ir* ai,i,)“’ and 1(A), the maximum cycle mean (MCM) of A, as max,,u(o), where o ranges over all cyclic permutations of subsets of N. We study the dependence of the MCM of a matrix on the permutations of its rows and columns and particularly we prove an O(n*) algorithm for checking whether 1(A) = rl(A’) holds for any matrix A’ which can be obtained from A by permuting its rows and columns.