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Complex data sets are often unmanageable unless they can be subdivided and simplified in an intelligent manner. Clustering is a technique that is used in data mining

This paper makes three contributions to cyber-security research. First, we define a model for cyber-security systems and the concept of a cyber-security attack within the model's framework. The model highlights the importance of game-over components-critical system components which if acquired will give an adversary the ability to defeat a system completely. The model is based on systems that use defense-in-depth/layered-security approaches, as many systems do. In the model we define the concept of penetration cost, which is the cost that must be paid in order to break into the next layer of security. Second, we define natural decision and optimization problems based on cyber-security attacks in terms of doubly weighted trees, and analyze their complexity. More precisely, given a tree T rooted at a vertex r, a penetrating cost edge function c on T , a target-acquisition vertex function p on T , the attacker's budget and the game-over threshold B, G ∈ Q + respectively, we consider the problem of determining the existence of a rooted subtree T ′ of T within the attacker's budget (that is, the sum of the costs of the edges in T ′ is less than or equal to B) with total acquisition value more than the game-over threshold (that is, the sum of the target values of the nodes in T ′ is greater than or equal to G). We prove that the general version of this problem is intractable, but does admit a polynomial time approximation scheme. We also analyze the complexity of three restricted versions of the problems, where the penetration cost is the constant function, integer-valued, and rational-valued among a given fixed number of distinct values. Using recursion and dynamicprogramming techniques, we show that for constant penetration costs an optimal cyber-attack strategy can be found in polynomial time, and for integer-valued and rational-valued penetration costs optimal cyber-attack strategies can be found in pseudo-polynomial time. Third, we provide a list of open problems relating to the architectural design of cyber-security systems and to the model.

We study the Graph Relabeling Problem-given an undirected, connected, simple graph G = (V, E), two labelings L and L ′ of G, and label flip or mutation functions determine the complexity of transforming or evolving the labeling L into L ′ . The transformation of L into L ′ can be viewed as an evolutionary process governed by the types of flips or mutations allowed. The number of applications of the function is the duration of the evolutionary period. The labels may reside on the vertices or the edges. We prove that vertex and edge relabelings have closely related computational complexities. Upper and lower bounds on the number of mutations required to evolve one labeling into another in a general graph are given. Exact bounds for the number of mutations required to evolve paths and stars are given. This corresponds to computing the exact distance between two vertices in the corresponding Cayley graph. We finally explore both vertex and edge relabeling with privileged labels, and resolve some open problems by providing precise characterizations of when these problems are solvable. Many of our results include algorithms for solving the problems, and in all cases the algorithms are polynomial-time. The problems studied have applications in areas such as bioinformatics, networks, and VLSI.

This paper makes three contributions to cyber-security research. First, we define a model for cyber-security systems and the concept of a cyber-security attack within the model's framework. The model highlights the importance of game-over components-critical system components which if acquired will give an adversary the ability to defeat a system completely. The model is based on systems that use defense-in-depth/layered-security approaches, as many systems do. In the model we define the concept of penetration cost, which is the cost that must be paid in order to break into the next layer of security. Second, we define natural decision and optimization problems based on cyber-security attacks in terms of doubly weighted trees, and analyze their complexity. More precisely, given a tree T rooted at a vertex r, a penetrating cost edge function c on T , a target-acquisition vertex function p on T , the attacker's budget and the game-over threshold B, G ∈ Q + respectively, we consider the problem of determining the existence of a rooted subtree T of T within the attacker's budget (that is, the sum of the costs of the edges in T is less than or equal to B) with total acquisition value more than the game-over threshold (that is, the sum of the target values of the nodes in T is greater than or equal to G). We prove that the general version of this problem is intractable, but does admit a polynomial time approximation scheme. We also analyze the complexity of three restricted versions of the problems, where the penetration cost is the constant function, integer-valued, and rational-valued among a given fixed number of distinct values. Using recursion and dynamic-programming techniques, we show that for constant penetration costs an optimal cyber-attack strategy can be found in polynomial time, and for integer-valued and rational-valued penetration costs optimal cyber-attack strategies can be found in pseudo-polynomial time. Third, we provide a list of open problems relating to the architectural design of cybersecurity systems and to the model.