AI 中文总结
研究双层优化中领导者与追随者联合构建组合优化可行解的分区项目响应问题，通过不同项目组合探讨可处理性，以最大权重匹配和最小权重完美匹配为例，给出多种情况下问题的难度及可处理性结论。
AI 中文摘要
我们考虑双层优化问题，其中领导者和追随者共同为一个潜在的组合优化问题构建可行解。响应问题询问领导者能否鼓励（在悲观情况下是强制）追随者做出一种反应，该反应包含一组强制项目同时排除一组禁止项目。我们的研究聚焦于由强制、禁止和中性项目总数的不同组合产生的各种情况的可处理性结果。在给出一些适用于任何潜在组合优化问题的响应问题结果后，我们将最大权重匹配问题和最小权重完美匹配问题作为示例进行研究，它们出人意料地具有多样性。我们还展示了即使只有一条给定的强制或禁止边，响应问题也很难。另一方面，它对于非强制边的总数是固定参数可处理的。然而，如果每个追随者的边要么是强制的要么是禁止的，那么完美匹配问题的响应问题在多项式时间内可解，而最大权重匹配问题的响应问题仍然是NP难的。
英文摘要
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.
