In the Stable Roommates Problem (SR), a set of $2n$ agents rank one another in a linear order. The goal is to find a matching that is stable: one that has no pair of agents who mutually prefer each other over their assigned partners. We consider the problem of finding an optimal stable matching. Agents associate weights with each of their potential partners, and the goal is to find a stable matching that minimizes the sum of the associated weights. Efficient algorithms exist for finding an optimal stable matching in the Stable Marriage Problem (SM), but the problem is NP-hard for general SR instances. In this paper, we define a notion of structural distance between SR instances and SM instances, which we call the minimum crossing distance. When an SR instance has minimum crossing distance $0$, the instance is structurally equivalent to an SM instance, and this structure can be exploited to find an optimal stable matching efficiently. More generally, we show that when an SR instance has minimum crossing distance $k$, an optimal stable matching can be computed in time $2^{O(k)} n^{O(1)}$. Thus, the optimal stable matching problem is fixed-parameter tractable (FPT) with respect to minimum crossing distance.

翻译：在稳定室友问题（SR）中，一组包含 $2n$ 个代理以线性顺序相互排序。目标是找到一个稳定的匹配：即不存在一对相互偏好的代理，且两者均更偏好对方而非当前分配伙伴。我们考虑寻找最优稳定匹配的问题。代理为每个潜在伙伴分配权重，目标是最小化所有权重之和的稳定匹配。尽管稳定婚姻问题（SM）中存在高效算法求解最优稳定匹配，但对于一般SR实例而言，该问题是NP难问题。在本文中，我们定义了SR实例与SM实例之间的一种结构距离概念，称为最小交叉距离。当SR实例的最小交叉距离为 $0$ 时，该实例在结构上等价于SM实例，并且可以利用这种结构高效地找到最优稳定匹配。更一般地，我们证明当SR实例的最小交叉距离为 $k$ 时，可以在 $2^{O(k)} n^{O(1)}$ 时间内计算出最优稳定匹配。因此，最优稳定匹配问题关于最小交叉距离是固定参数可处理的（FPT）。