The computational complexity of bilevel assignment problems

In bilevel optimization problems there are two decision makers, the leader and the follower, who act in a hierarchy. Each decision maker has his own objective function, but there are common constraints. This paper deals with bilevel assignment problems where each decision maker controls a subset of edges and each edge has a leader’s and a follower’s weight. The edges selected by the leader and by the follower need to form a perfect matching. The task is to determine which edges the leader should choose such that his objective value which depends on the follower’s optimal reaction is maximized. We consider sum- and bottleneck objective functions for the leader and follower. Moreover, if not all optimal reactions of the follower lead to the same leader’s objective value, then the follower either chooses an optimal reaction which is best (optimistic rule) or worst (pessimistic rule) for the leader. We show that all the variants arising if the leader’s and follower’s objective functions are sum or bottleneck functions are NP-hard if the pessimistic rule is applied. In case of the optimistic rule the problem is shown to be NP-hard if at least one of the decision makers has a sum objective function.