Privacy preserving clustering with constraints

The \(k\)-center problem is a classical combinatorial optimization problem which asks to find \(k\) centers such that the maximum distance of any input point in a set \(P\) to its assigned center is minimized. The problem allows for elegant \(2\)-approximations. However, the situation becomes significantly more difficult when constraints are added to the problem. We raise the question whether general methods can be derived to turn an approximation algorithm for a clustering problem with some constraints into an approximation algorithm that respects one constraint more. Our constraint of choice is privacy: Here, we are asked to only open a center when at least \(\ell\) clients will be assigned to it. We show how to combine privacy with several other constraints.