Critical edges for the assignment problem: Complexity and exact resolution

This paper investigates two problems related to the determination of critical edges for the minimum cost assignment problem. Given a complete bipartite balanced graph with n vertices on each part and with costs on its edges, k Most Vital Edges Assignment consists of determining a set of k edges whose removal results in the largest increase in the cost of a minimum cost assignment. A dual problem, Min Edge Blocker Assignment, consists of removing a subset of edges of minimum cardinality such that the cost of a minimum cost assignment in the remaining graph is larger than or equal to a specified threshold. We show that k Most Vital Edges Assignment is N P -hard to approximate within a factor c < 2 and Min Edge Blocker Assignment is N P -hard to approximate within a factor 1.36 . We also provide an exact algorithm for k Most Vital Edges Assignment that runs in O ( n k + 2 ) . This algorithm can also be used to solve exactly Min Edge Blocker Assignment.