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 {\it 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 optimal stable marriages 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 \emph{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 optimal stable matchings efficiently. More generally, we show that for an SR instance with minimum crossing distance $k$, optimal stable matchings 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 $2n$ 个代理按线性顺序相互排名，目标是找到一个不存在相互偏好对方胜过当前伴侣的配对对的稳定匹配。我们考虑寻找{\it最优}稳定匹配的问题：代理为其每个潜在伴侣赋予权重，目标是最小化所有权重之和的稳定匹配。稳定婚姻问题（SM）中已存在寻找最优稳定婚姻的高效算法，但一般SR实例下的该问题是NP难的。本文定义了SR实例与SM实例之间的结构距离概念，称为\emph{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，且可利用此结构高效求解最优稳定匹配。更一般地，我们证明：对于最小交叉距离为$k$的SR实例，可在$2^{O(k)} n^{O(1)}$时间内计算出最优稳定匹配。因此，最优稳定匹配问题相对于最小交叉距离是固定参数可解的（FPT）。