In this paper we consider strategic cost sharing games with so-called arbitrary sharing based on various combinatorial optimization problems, such as vertex and set cover, facility location, and network design problems. We concentrate on the existence and computational complexity of strong equilibria, in which no coalition can improve the cost of each of its members. Our main result reveals a connection between strong equilibrium in strategic games and the core in traditional coalitional cost sharing games studied in economics. For set cover and facility location games this results in a tight characterization of the existence of strong equilibrium using the integrality gap of suitable linear programming formulations. Furthermore, it allows to derive all existing results for strong equilibria in network design cost sharing games with arbitrary sharing via a unified approach. In addition, we are able to show that in general the strong price of anarchy is always 1. This should be contrasted with the price of anarchy of Θ(n) for Nash equilibria. Finally, we indicate that the LP-approach can also be used to compute near-optimal and near-stable approximate strong equilibria.