🤖 AI Summary
This study addresses the feasibility of leader-induced follower responses in bilevel matching optimization, where the leader specifies mandatory and forbidden edges. Focusing on maximum-weight matchings and minimum-weight perfect matchings, the work combines parameterized complexity analysis, combinatorial optimization theory, and bilevel modeling to show that even with a single mandatory or forbidden edge, the follower response problem for general matchings remains NP-hard. In contrast, when every edge is either mandatory or forbidden, the perfect matching variant becomes polynomial-time solvable. Furthermore, the problem is shown to be fixed-parameter tractable with respect to the number of non-mandatory edges, thereby precisely delineating the complexity boundary between these two matching settings under structural constraints.
📝 Abstract
We consider bilevel optimization problems in which leader and follower jointly construct a feasible solution for an underlying combinatorial optimization problem. Response problems ask whether the leader can encourage -- or, in the pessimistic setting, enforce -- a reaction of the follower that includes a set of mandatory items while excluding a set of forbidden items. Our investigation focuses on tractability results for various cases which emerge from different combinations of the total number of mandatory, forbidden, and neutral items. After providing some results for response problems that hold for any underlying combinatorial optimization problem, we examine response problems over the maximum-weight matching problem and the minimum-weight perfect matching problem as illustrative and surprisingly varied examples. Among other results, we show that the response problem is hard for even a single given mandatory or forbidden edge. On the other hand, it is fixed-parameter tractable with respect to the total number of non-mandatory edges. If, however, each follower's edge is either mandatory or forbidden, the response problem for the perfect matching problem is solvable in polynomial time while it remains NP-hard for the maximum-weight matching problem.