The stable path problem (SPP) is a unified model for analyzing the convergence of distributed routing protocols (e.g., BGP), and a foundation for many network verification tools. Although substantial progress has been made on finding solutions (i.e., stable path assignments) for particular subclasses of SPP instances and analyzing the relation between properties of SPP instances and the convergence of corresponding routing policies, the non-trivial challenge of finding stable path assignments to generic SPP instances still remains. Tackling this challenge is important because it can enable multiple important, novel routing use cases. To fill this gap, in this paper we introduce a novel data structure called solvability digraph, which encodes key properties about stable path assignments in a compact graph representation. Thus SPP is equivalently transformed to the problem of finding in the solvability digraph a maximum independent set (MIS) of size equal to the number of autonomous systems (ASes) in the given SPP instance. We leverage this key finding to develop a heuristic polynomial algorithm GREEDYMIS that solves strictly more SPP instances than state-of-the-art heuristics. We apply GREEDYMIS to designing two important, novel use cases: (1) a centralized interdomain routing system that uses GREEDYMIS to compute paths for ASes and (2) a secure multi-party computation (SMPC) protocol that allows ASes to use GREEDYMIS collaboratively to compute paths without exposing their routing preferences. We demonstrate the benefits and efficiency of these use cases via evaluation using real-world datasets.