  The $k$-cardinality assignment problem asks for finding a maximal (minimal)
weight of a matching of cardinality $k$ in a weighted bipartite graph
$K_{n,n}$, $k \leq n$. The algorithm of Gassner and Klinz from 2010 for the
parametric assignment problem computes in time $O(n^3)$ the set of
$k$-cardinality assignments for those integers $k \leq n$ which refer to
"essential" terms of a corresponding maxpolynomial. We show here that one can
extend this algorithm and compute in a second stage the other "semi-essential"
terms in time $O(n^2)$, which results in a time complexity of $O(n^3)$ for the
whole sequence of $k=1,...,n$-cardinality assignments. The more there are
assignments left to be computed at the second stage the faster the two-stage
algorithm runs. In general, however, there is no benefit for this two-stage
algorithm on the existing algorithms, e.g. the simpler network flow algorithm
based on the successive shortest path algorithm which also computes all the
$k$-cardinality assignments in time $O(n^3)$.
