We study the Reaching Stable Marriage via Divorces (DivorceSM) problem of deciding, given a Stable Marriage instance and an initial matching $M$ , whether there exists a stable matching which is reachable from $M$ by divorce operations as introduced by Knuth [12]. Towards answering an open question of Manlove [13] and Cechl\'arov\'a et al. [3], we show that for incomplete preferences without ties, DivorceSM is NP-hard. Our hardness reduction also implies that the problem remains parameterized intractable for the number $\kappa$ of allowed divorce operations. It remains NP-hard even if the maximum length $d$ of the preferences is a constant. For the combined parameter ($\kappa$, $d$), the problem is fixed-parameter tractable.

翻译：我们研究了通过离婚实现稳定的婚姻的问题,考虑到一个稳定的婚姻案例和初步匹配的美元,确定是否存在一个稳定匹配,Knuth[12] 提出的离婚行动能否从M$获得。为了回答Manlove[13]和Cechl\'arov\a'a et al.[3]的未决问题,我们发现,对于没有联系的不完全的偏好,离婚SM是NP-硬的。我们的硬性降低还意味着,对于允许离婚业务的美元数来说,问题仍然是参数化的棘手问题。即使最长时间的优惠额是不变的,但NP仍然是硬的。对于综合参数(kappa$,$d$)来说,问题是可以固定的。