Critical Infrastructures like power and communication networks are highly interdependent on each other for their full functionality. Many significant research have been pursued to model the interdependency and failure analysis of these interdependent networks. However, most of these models fail to capture the complex interdependencies that might actually exist between the infrastructures. The \emph{Implicative Interdependency Model} that utilizes Boolean Logic to capture complex interdependencies was recently proposed which overcome the limitations of the existing models. A number of problems were studies based on this model. In this paper we study the \textit{Robustness} problem in Interdependent Power and Communication Network. The robustness is defined with respect to two parameters $K \in I^{+} \cup \{0\}$ and $\rho \in (0,1]$. We utilized the \emph{Implicative Interdependency Model} model to capture the complex interdependency between the two networks. The model classifies the interdependency relations into four cases. Computational complexity of the problem is analyzed for each of these cases. A polynomial time algorithm is designed for the first case that outputs the optimal solution. All the other cases are proved to be NP-complete. An in-approximability bound is provided for the third case. For the general case we formulate an Integer Linear Program to get the optimal solution and a polynomial time heuristic. The applicability of the heuristic is evaluated using power and communication network data of Maricopa County, Arizona. The experimental results showed that the heuristic almost always produced near optimal value of parameter $K$ for $\rho < 0.42$.
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