Bipartite Complete Matching Vertex Interdiction Problem: Application to Robust Nurse Assignment

In this paper, we consider the Robust Nurse Assignment Problem. This consists in finding the maximum number of absences of qualified nurses still permitting an optimal treatment of patients, leading us to the notion of critical jobs. We introduce the Bipartite Complete Matching Vertex Interdiction Problem as the graph formulation of this problem. We show that it can be solved in polynomial time thanks to the integer polytope of an associated sub-problem. Then, we study the polytope associated with the Bipartite Complete Matching Vertex Interdiction Problem. We also extend the well-known Hall theorem to this problem.