Computing the sequence of k-cardinality assignments
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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 ≤ 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 ≤ 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).
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