Properties of Some Extremal Problems of Permutation Cycles

Given a set of n real numbers A = {ai|i = 1, …, n} and a permutation α of the integers from 1 to n, define the function f(α, A) = 1/2 Σin=1(aαi - aαi+1)2. We determine the permutations α* and α** which maximize and minimize this function. Next, given the integer k, we find the subsets Bk* and Bk** which maximize and minimize f(α*, Bk*) and f(α**, Bk**) respectively, over all subsets Bk of A with cardinality k. Then we determine the values for k, 2⩽k⩽n which maximize and minimize the values of f(α*, Bk*) and f(α**, Bk**) respectively. These results are then applied to a special assignment problem to determine if the diameter of two property of the assignment can be achieved by an adjacent extreme point method, i.e. can the problem be solved in two steps. It is shown that in general this is not possible.