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In this paper, we introduce the Detachment Problem. It can be seen as ageneralized Vaccination Problem. The aim is to optimally cut the individuals'ties to circles that connect them to others, to minimize the overallinformation transfer in a social network. When an individual is isolated from aparticular circle, it leads to the elimination of the connections to all themembers of that circle, yet the connections to other circles remain. Thisapproach contrasts with the conventional vaccination problem, in which a subsetof vertices is totally eliminated. In our case, the connections of individualsto their circles are selectively, rather than entirely, eliminated.Contextually, this article focuses on private information flows, specificallywithin networks formed by memberships in circles of insiders in companies. Ourquasi-empirical study uses simulated information flows on an observablenetwork, and the statistical properties of the simulated information flows arematched with real-world data. In a broader context, this paper presents theDetachment Problem as a versatile approach for optimal social distancing,applicable across various scenarios. We propose and define a concept ofexpected proportional outside influence, or EPOI, as measure of how widespreadinformation leak is. We also implement a greedy algorithm for finding a set ofdetachments to minimize EPOI. For comparison, we devise a simple heuristicbased on minimal cut, to separate the most influential circles from each other.We provide evidence that the greedy algorithm is not optimal, and it issometimes outperformed by the simple heuristic minimum cut algorithm, However,the greedy algorithm outperforms the cut algorithm in most cases. Furtheravenues of research are discussed.

Detachment Problem -- Application in Prevention of Information Leakage in Stock Markets