BGP, the core protocol of the Internet backbone, is renowned to be prone to oscillations. Despite prior work shed some light on BGP stability, many problems remain open. For example, determining how hard it is to check that a BGP network is safe, i.e., it is guaranteed to converge, has been an elusive research goal up to now. In this paper, we address several problems related to BGP stability, stating the computational complexity of testing if a given configuration is safe, is robust, or is safe under filtering. Further, we determine the computational complexity of checking popular sufficient conditions for stability. We adopt a model that captures Local Transit policies, i.e., policies that are functions only of the ingress and the egress points. The focus on Local Transit policies is motivated by the fact that they represent a configuration paradigm commonly used by network operators. We also address the same BGP stability problems in the widely adopted SPP model. Unfortunately, we find that the most interesting problems are computationally hard even if policies are restricted to be as expressive as Local Transit policies. Our findings suggest that the computational intractability of BGP stability be an intrinsic property of policy-based path vector routing protocols that allow policies to be specified in complete autonomy.