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