Before revision:  We characterize the optimal protection problem  as a variant Steiner tree problem in a graph , and propose both exact and reduced-complexity approximation algorithms to derive the optimal protection strategy that achieves the protection objective with minimum cost.
After revision:  We characterize the optimal protection problem as a variant Steiner tree problem in a graph . Based on the graphical characterization, we propose both exact and reduced-complexity approximation algorithms to derive the optimal protection strategy that achieves the protection objective with minimum cost.