Balancing profits and costs on trees

Coene, Sofie; Filippi, Carlo; Spieksma, Frits; Stevanato, Elisa

doi:10.1002/net.21469

Cited by 14 publications

(16 citation statements)

References 36 publications

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“…Observation 3.1 answers an open question in the last section of [15], where it is asked whether or not the LST-Tree Problem can be solved in polynomial time (we presume) for general edge lengths. Observation 3.1 is similar to [7,Theorem 2], where also a star is considered to show that their SubtreeE is as hard as Knapsack.…”

Section: Complexity Of Cyber-attack Problemsmentioning

confidence: 55%

“…Also, the results of fixed costs, in our case Theorem 4.1 and in their case [7,Theorem 3], the problems are shown to be solvable in O(n) time, given certain conditions. Theorem 4.1, however, provides a precise accounting for the time complexity and for certain values of m, defined there, our algorithm would be faster than that given in [7]. Their work is not in the context of cyber-security, and does not handle cases as general as this work.…”

Section: Cyber Attacks With Constant Penetration Costsmentioning

confidence: 70%

“…Both here and in [7] the most general form of the problems considered, in our case GOAS-DP in Observation 3.1 and in their case (as mentioned above) SubtreeE in [7,Theorem 2], are observed to be as hard as Knapsack and hence NP-complete. Also, the results of fixed costs, in our case Theorem 4.1 and in their case [7,Theorem 3], the problems are shown to be solvable in O(n) time, given certain conditions. Theorem 4.1, however, provides a precise accounting for the time complexity and for certain values of m, defined there, our algorithm would be faster than that given in [7].…”

Section: Cyber Attacks With Constant Penetration Costsmentioning

confidence: 78%

See 2 more Smart Citations

On Cyber Attacks and the Maximum-Weight Rooted-Subtree Problem

2016

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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.

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Section: Complexity Of Cyber-attack Problemsmentioning

confidence: 55%

Section: Cyber Attacks With Constant Penetration Costsmentioning

confidence: 70%

Section: Cyber Attacks With Constant Penetration Costsmentioning

confidence: 78%

See 1 more Smart Citation

On Cyber Attacks and the Maximum-Weight Rooted-Subtree Problem

2016

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“…Remark: Observation 3.1 answers an open question in the last section of [15], where it is asked whether or not the LST-Tree Problem can be solved in polynomial time (we presume) for general edge lengths. Observation 3.1 is similar to [7,Theorem 2], where also a star is considered to show that their SubtreeE is as hard as Knapsack.…”

Section: And Vice Versa Each Collection Of Verticesmentioning

confidence: 55%

The Complexity of Cyber Attacks in a New Layered-Security Model and the Maximum-Weight, Rooted-Subtree Problem

Agnarsson

Greenlaw

Kantabutra

2015

New Information and Communication Technologies for Knowledge Management in Organizations

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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.

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“…As a result, quite a bit of work has been done where cybersecurity systems, or more generally layered computer systems, are modeled as a fixed weighted trees. For example, in [1,3,4,8,10,12] the authors consider finding weight-constrained, maximum-density subtrees and similar structures given a fixed weighting of a tree as part of the input. In these cases weights are specified on both vertices and edges.…”

Section: Introductionmentioning

confidence: 99%

The Structure of Rooted Weighted Trees Modeling Layered Cyber-security Systems

Agnarsson¹,

Greenlaw²,

Kantabutra³

2016

Acta Cybern

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In this paper we consider the structure and topology of a layered-security model in which the containers and their nestings are given in the form of a rooted tree T . A cyber-security model is an ordered three-tuple M = (T, C, P ) where C and P are multisets of penetration costs for the containers and targetacquisition values for the prizes that are located within the containers, respectively, both of the same cardinality as the set of the non-root vertices of T . The problem that we study is to assign the penetration costs to the edges and the target-acquisition values to the vertices of the tree T in such a way that minimizes the total prize that an attacker can acquire given a limited budget. The attacker breaks into containers starting at the root of T and once a vertex has been broken into, its children can be broken into by paying the associated penetration costs. The attacker must deduct the corresponding penetration cost from the budget, as each new container is broken into. For a given assignment of costs and target values we obtain a security system. We show that in general it is not possible to develop an optimal security system for a given cyber-security model M . We define P-and C-models where the penetration costs and prizes, respectively, all have unit value. We show that if T is a rooted tree such that any P-or C-model M = (T, C, P ) has an optimal security system, then T is one of the following types: (i) a rooted path, (ii) a rooted star, (iii) a rooted 3-caterpillar, or (iv) a rooted 4-spider. Conversely, if T is one of these four types of trees, then we show that any Por C-model M = (T, C, P ) does have an optimal security system. Finally, we study a duality between P-and C-models that allows us to translate results for P-models into corresponding results for C-models and vice versa. The results obtained give us some mathematical insights into how layered-security defenses should be organized.

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Balancing profits and costs on trees

Cited by 14 publications

References 36 publications

On Cyber Attacks and the Maximum-Weight Rooted-Subtree Problem

On Cyber Attacks and the Maximum-Weight Rooted-Subtree Problem

The Complexity of Cyber Attacks in a New Layered-Security Model and the Maximum-Weight, Rooted-Subtree Problem

The Structure of Rooted Weighted Trees Modeling Layered Cyber-security Systems

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