In this paper, we introduce the Detachment Problem. It can be seen as a generalized Vaccination Problem. The aim is to optimally cut the individuals' ties to circles that connect them to others, to minimize the overall information transfer in a social network. When an individual is isolated from a particular circle, it leads to the elimination of the connections to all the members of that circle, yet the connections to other circles remain. This approach contrasts with the conventional vaccination problem, in which a subset of vertices is totally eliminated. In our case, the connections of individuals to their circles are selectively, rather than entirely, eliminated. Contextually, this article focuses on private information flows, specifically within networks formed by memberships in circles of insiders in companies. Our quasi-empirical study uses simulated information flows on an observable network, and the statistical properties of the simulated information flows are matched with real-world data. In a broader context, this paper presents the Detachment Problem as a versatile approach for optimal social distancing, applicable across various scenarios. We propose and define a concept of expected proportional outside influence, or EPOI, as measure of how widespread information leak is. We also implement a greedy algorithm for finding a set of detachments to minimize EPOI. For comparison, we devise a simple heuristic based on minimal cut, to separate the most influential circles from each other. We provide evidence that the greedy algorithm is not optimal, and it is sometimes outperformed by the simple heuristic minimum cut algorithm, However, the greedy algorithm outperforms the cut algorithm in most cases. Further avenues of research are discussed.