We consider a generalization of the assignment problem in which an integer k is given and one wants to assign k rows to k columns so that the sum of the corresponding costs is a minimum. The problem can be seen as a 2-matroid intersection, hence is solvable in polynomial time; immediate algorithms for it can be obtained from transformation to min-cost flow or from classical shortest augmenting path techniques. We introduce original preprocessing techniques for finding optimal solutions in which g≤k rows are assigned, for determining rows and columns which must be assigned in an optimal solution and for reducing the cost matrix. A specialized primal algorithm is finally presented. The average computational efficiency of the different approaches is evaluated through computational experiments.
Dell’Amico, M., & Martello, S. (1997). The k-cardinality assignment problem. Discrete Applied Mathematics, 76(1–3), 103–121. https://doi.org/10.1016/S0166-218X(97)00120-0