Minimizing maximum weight of subsets of a maximum matching in a bipartite graph

In a complete bipartite graph G = ( U , V , E ) with weighted edges, set U of vertices is partitioned into disjoint subsets called components. The aim is to find an inclusion-wise maximal matching that minimizes the maximum weight of a component (sum of the weights of those edges incident to a vertex of the component). The well-known assignment problem is a special case of this problem when the number of components equals one. We prove that the problem is strongly NP-hard and that finding an approximate solution with the approximation ratio less than 2 is strongly NP-hard as well. We develop heuristics to find solutions that are less than one percent away from the optimum on average. The problem has been encountered while planning container transportation in rail-road transshipment yards.