The IE algorithm works by restricting and subsequently relaxing the defense budget of the original tri-level DAD problem in order to generate easier to calculate bi- level AD sub problems that can help us find all reasonable choices of defenses. The restriction of the problem is performed by fixing the values of the defense variables (Wˆ ) one at a time. At the root node, the defender is restricted in having zero defense budget, which is the initial AD sub problem. At the first level of depth, the restriction of the root node problem is eased slightly because the defender is allowed one defense. At depth one
in the IE tree, the defender can choose only one defense that either improves the objective function or changes the worst-case attack of the parent node AD sub problem. This rule for node development at the next depth effectively prunes all other possible defenses that would be suboptimal. In this manner, explicit enumeration is avoided because all possible choices of defenses are not calculated in the tree. All of the AD subproblem instances in the IE tree are also easier to calculate than the original DAD problem instance. The methodology of preventing repeat calculations also helps to prune repeat calculations from the enumeration tree. The process of slight easing of the restriction on fixing the defense variables will continue throughout the formation of the IE tree until the defense budget is exhausted. Additionally, the formation of each new node performs a check for the updating of the best known feasible solution. This feature allows the analyst to be able to pause the algorithm and get the best known feasible solution at any time.
Proposition 2–9: The IE algorithm is an exact algorithm that produces an optimal solution to a DAD problem instance. If the IE tree is completely explored, then one or more equivalent optimal solutions will be found to the DAD problem instance.
Proof of proposition 2–9: An enumeration strategy lists all possible solutions to a problem instance. IE discards feasible combinations of decision variables that cannot lead to an optimal solution. In our implementation of IE for a DAD problem instance, we compute only those feasible combinations of defenses to worst-case attacks that can improve the objective function value or change the worst-case attack from a previous AD subproblem instance. The complete IE tree will list all feasible combinations of defenses that can either improve the objective function value or change worst-case attacks. The combinations of defenses that do not do either of those two things cannot lead to an optimal solution to the DAD problem instance. If the tree exploration is prematurely halted before completing the IE algorithm, an optimal solution might be found, but it also may exist in a portion of the tree that had not yet been completely explored.▪
However, IE does have a shortcoming. The size of an enumeration tree depends on the magnitude of the defense and attack budgets, the costs of defenses and attacks to those budgets, and the cardinality of the set of new edges that may be added to the network. The depth of the tree is equal to the number of defenses selected in AD sub
problems. The breadth of the tree at any particular depth depends upon the number of attacks permitted by the attacker budget and the cardinality of the set of new edges. Thus, the shortcoming of IE is that the tree size can become incredibly large when either the defense budget, attack budget, or number of network design elements becomes large.
In the case where there are no new edges that can be added to the network, the maximum size of the tree is straightforward to compute. The breadth of each level of the tree is the number of attacks raised to the exponent of the level of the tree. Thus, with three defenses and two attacks, the maximum size of the tree is 15 nodes. The maximum size is computed as follows: level zero consists of 20=1 node; level one has 21=2 nodes; level two has 22=4 nodes; level three contains 23=8 nodes. The maximum number of enumeration nodes in this tree is sum of all levels, 20 +21+22+ 23 = 15 nodes. The tree gets really big fast. If there were 5 attacks and 5 defenses, the maximum number of enumeration nodes would be 50+51+52+53+54+55 = 3906 enumeration nodes, each containing an AD mixed integer program (MIP) subproblem. However, the maximum size of the tree is not the actual size of the tree when exclusion lists are used to prevent repeated calculations. Nevertheless, we observe the problematic issue with IE is the number of MIPs to solve grows exponentially with the number of defenses allowed by the defense budget. Adding new edges into consideration makes the maximum number of nodes even larger.
IE may not always be the fastest way to the optimal solution to a DAD problem instance, but this approach does have the unique capability of quickly identifying multiple optimal solutions. If an analyst needs to consider all equivalent optimal solutions in order to make a list of defenses to choose, IE may be a good choice.