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 \(K_{n,n}\), \(k \le n\). Here we are interested in computing the sequence of all k-assignments, \(k=1,\ldots ,n\). By applying the algorithm of Gassner and Klinz (2010) for the parametric assignment problem one can compute in time \({\mathcal {O}}(n^3)\) the set of k-assignments for those integers \(k \le n\) which refer to essential terms of the full characteristic maxpolynomial \({\bar{\chi }}_{W}(x)\) of the corresponding max-plus weight matrix W. We show that \({\bar{\chi }}_{W}(x)\) is in full canonical form, which implies that the remaining k-assignments refer to semi-essential terms of \({\bar{\chi }}_{W}(x)\). This property enables us to efficiently compute in time \({\mathcal {O}}(n^2)\) 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 \({\mathcal {O}}(n^3)\), which is the best known time for this problem.

k -基数分配（简称 k -assignment ）问题要求在加权二分图中找到基数k匹配的最小（最大）权重\(K_{n,n}\) , \(k \le n\)。在这里，我们感兴趣的是计算所有k赋值的序列\(k=1,\ldots ,n\)。通过应用 Gassner 和 Klinz (2010) 的算法来解决参数分配问题，我们可以及时计算\({\mathcal {O}}(n^3)\)这些整数的k个分配集合\(k \ le n\)指完整特征最大多项式的基本项\({\bar{\chi }}_{W}(x)\)对应的最大加权重矩阵W。我们证明\({\bar{\chi }}_{W}(x)\)是完全规范的形式，这意味着剩余的k赋值指的是\({\bar{\智 }}_{W}(x)\)。该属性使我们能够及时有效地计算\({\mathcal {O}}(n^2)\)已经计算的基本k赋值中的所有剩余k赋值。因此，计算所有k基数分配序列的时间复杂度为\({\mathcal {O}}(n^3)\)，这是该问题的最知名时间。