Polynomially solvable cases of the bipartite traveling salesman problem

Given two sets, R and B, consisting of n cities each, in the bipartite traveling salesman problem one looks for the shortest way of visiting alternately the cities of R and B, returning to the city of origin. This problem is known to be NP-hard for arbitrary sets R and B. In this paper we provide an O(n6) algorithm to solve the bipartite traveling salesman problem if the quadrangle property holds. In particular, this algorithm can be applied to solve in O(n6) time the bipartite traveling salesman problem in the following cases: S=R¿B is a convex point set in the plane, S=R¿B is the set of vertices of a simple polygon and V=R¿B is the set of vertices of a circular graph. For this last case, we also describe another algorithm which runs in O(n2) time

New special cases of the quadratic assignment problem with diagonally structured coefficient matrices

We consider new polynomially solvable cases of the well-known Quadratic Assignment Problem involving coefficient matrices with a special diagonal structure. By combining the new special cases with polynomially solvable special cases known in the literature we obtain a new and larger class of polynomially solvable special cases of the QAP where one of the two coefficient matrices involved is a Robinson matrix with an additional structural property: this matrix can be represented as a conic combination of cut matrices in a certain normal form. The other matrix is a conic combination of a monotone anti-Monge matrix and a down-benevolent Toeplitz matrix. We consider the recognition problem for the special class of Robinson matrices mentioned above and show that it can be solved in polynomial time

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

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 ϵ 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 cyber-security systems and to the model

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

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 target-acquisition 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 P- or 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