Computing the sequence of k-cardinality assignments

The k-cardinality assignment (k-assignment, for short) problem asks for finding a minimal (maximal) weight of a matching of cardinality k in a weighted bipartite graph K_n,n , k ≤ n. Here we are interested in computing the sequence of all k-assignments, k=1,...,n. By applying the algorithm of Gassner and Klinz (2010) for the parametric assignment problem one can compute in time O(n³) the set of k-assignments for those integers k≤n which refer to essential terms of the full characteristic maxpolynomial χ_W(x) of the corresponding max-plus weight matrix W. We show that χ_W(x) is in full canonical form, which implies that the remaining k-assignments refer to semi-essential terms of χ_W(x). This property enables us to efficiently compute in time O(n²)  all the remaining k-assignments out of the already computed essential k-assignments. It follows that time complexity for computing the sequence of all k-cardinality assignments is O(n³) ,which is the best known time for this problem.}