## 4 Communication Network Interdiction  
An important issue in military applications is to neutralize the communication in the sensors network of the enemy. This problem is known as jamming or eavesdropping a wireless communication network. This section introduces optimization formulations that allow to place jamming devices delivering maximal harm to the adverse sensors network. We start from a deterministic case when node locations are known. The goal of jamming is to find a set of locations for placing jamming devices that suppresses the functionality of the network. Assume that \(n\) jamming devices are used to jam \(m\) communicating sensors. The assumption is made that the sensors and jammers can be located on a fixed set of locations \(V\). The jamming effectiveness of device \(j\) is calculated as  
\[d:(V\times V)\mapsto I\!\!R,\]  
where \(d\) is a decreasing function of the distance from the jamming device to the node being jammed. The cumulative level of jamming energy received at node \(i\) is defined as  
\[Q_{i}:=\sum_{j=1}^{n}d_{ij},\]  
where \(n\) is the number of jamming devices. Then, jamming problem can be formulated as the minimization of the number of jamming devices placed, subject to a set of covering constraints:  
\[\min n\] \[\text{s.t.}Q_{i}\geq C_{i},\quad i=1,2,\ldots,m.\]  
Seeking the optimal placement coordinates \((x_{j},y_{j}),j=1,2,\ldots,n\) for jamming devices given the coordinates \((X_{i},Y_{i}),i=1,2,\ldots,m\) leads to non-convex formulations for most functions \(d\). Thus, integer programming models for the problem are proposed. A fixed set \(\mathcal{N}=\{1,2,\ldots,n\}\) of possible locations for the jamming devices and the set of communication nodes are introduced by [PERSON] _et al_ in [33]. Define the decision variable \(x_{j}\) as  
\[x_{j}= \begin{cases}1,\text{ if a jamming device is installed at location }j,\\ 0,\text{ otherwise}.\end{cases}  \tag{28}\]Then we have the optimal network covering formulation given as  
\[\min\sum_{j=1}^{n}c_{j}x_{j}\] (29) s.t. (30) \[\sum_{j=1}^{n}d_{ij}x_{j}\geq C_{i},\quad i=1,2,\ldots,m, \tag{31}\] \[x_{j}\in\{0,1\},\quad j=1,2,\ldots,n, \tag{32}\]  
Here the objective is to minimize the cost of jamming devices used while achieving some minimum level of coverage at each node. If \(c_{j}=1\) then the number of jammers is minimized. If the goal is to suppress sensors communications we can minimize jamming cost with respect to the required level of connectivity index. Communication between nodes \(i\) and \(j\) is assumed to be destroyed if at least one of the nodes is jammed. Further, let \(y_{ij}:=1\) if there exists a path from node \(i\) to node \(j\) in the jammed network and let \(z_{i}=1\) be an indicator that \(i\)-th node is jammed. This can be formulated as  
\[\min\sum_{j=1}^{n}c_{j}x_{j}\] (33) s.t. \[\sum_{ \begin{subarray}{c}j=1\\ j\
eq i\end{subarray} }^{m}y_{ij}\leq L,\;\forall\;i\in\mathcal{M}, \tag{34}\] \[M(1-z_{i})>S_{i}-C_{i}\geq-Mz_{i},\;\forall\;i\in\mathcal{M},\] (35) \[y_{ij}\text{ is consistent with the network and }z_{i}\] (36) \[x_{j}\in\{0,1\},\;\forall\;j\in\mathcal{N},\] (37) \[z_{i}\in\{0,1\},\;\forall\;i\in\mathcal{M},\] (38) \[y_{ij}\in\{0,1\},\;\forall\;i,j\in\mathcal{M}, \tag{39}\]  
where \(S_{i}:=\sum_{j=1}^{n}d_{ij}x_{j}\) denote the cumulative level of jamming at node \(i\), \(M\in I\!\!R\) is some large constant. This problem can be formulated as a mixed integer linear problem and justification is provided in [33]. Finally, [PERSON] _et al_ provide percentile-based formulation for deterministic jamming problems. Suppose it is determined that jamming some fraction \(\alpha\in(0,1)\) of the nodes is sufficient for effectively dismantling the network. This can be accomplished by the inclusion of \(\alpha\)-VaR constraints in the original model. Let \(y:\mathcal{M}\mapsto\{0,1\}\) be an indicator whether node \(i\) is jammed (\(y_{i}=1\)).