Computing the sequence of k-cardinality assignments

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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 Kn,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 kn which refer to essential terms of the full characteristic maxpolynomial χW(x) of the corresponding max-plus weight matrix W. We show that χW(x) is in full canonical form, which implies that the remaining k-assignments refer to semi-essential terms of χW(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.}
Original languageEnglish
Number of pages19
JournalJournal of combinatorial optimization
Publication statusE-pub ahead of print - 24 Jul 2022


  • k-cardinality assignment problem
  • max-plus algebra

ASJC Scopus subject areas

  • Control and Optimization


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