The linearization problem of a binary quadratic problem and its applications

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore-Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. Finally, we present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem

The Quadratic Shortest Path Problem and its Genetic Algorithm

The quadratic shortest path (QSP) problem is to find a path from a node to another node in a given network such that the total cost includes two kinds of costs, say direct cost and interactive cost, is minimum. The direct cost is the cost associated with each arc and the interactive cost occurs when two arcs appear simultaneously in the shortest path. In this paper, the concept of the quadratic shortest path is initialized firstly. Then a spanning tree-based genetic algorithm is designed for solving the quadratic shortest path problem. Finally, a numerical example is given

The Quadratic Cycle Cover Problem: special cases and efficient bounds

The quadratic cycle cover problem is the problem of finding a set of node-disjoint cycles visiting all the nodes such that the total sum of interaction costs between consecutive arcs is minimized. In this paper we study the linearization problem for the quadratic cycle cover problem and related lower bounds. In particular, we derive various sufficient conditions for the quadratic cost matrix to be linearizable, and use these conditions to compute bounds. We also show how to use a sufficient condition for linearizability within an iterative bounding procedure. In each step, our algorithm computes the best equivalent representation of the quadratic cost matrix and its optimal linearizable matrix with respect to the given sufficient condition for linearizability. Further, we show that the classical Gilmore-Lawler type bound belongs to the family of linearization based bounds, and therefore apply the above mentioned iterative reformulation technique. We also prove that the linearization vectors resulting from this iterative approach satisfy the constant value property. The best among here introduced bounds outperform existing lower bounds when taking both quality and efficiency into account

A Multi-Agent Systems Approach for Analysis of Stepping Stone Attacks

Stepping stone attacks are one of the most sophisticated cyber-attacks, in which attackers make a chain of compromised hosts to reach a victim target. In this Dissertation, an analytic model with Multi-Agent systems approach has been proposed to analyze the propagation of stepping stones attacks in dynamic vulnerability graphs. Because the vulnerability configuration in a network is inherently dynamic, in this Dissertation a biased min-consensus technique for dynamic graphs with fixed and switching topology is proposed as a distributed technique to calculate the most vulnerable path for stepping stones attacks in dynamic vulnerability graphs. We use min-plus algebra to analyze and provide necessary and sufficient convergence conditions to the shortest path in the fixed topology case. A necessary condition for the switching topology case is provided. Most cyber-attacks involve an attacker launching a multi-stage attack by exploiting a sequence of hosts. This multi-stage attack generates a chain of ``stepping stones” from the origin to target. The choice of stepping stones is a function of the degree of exploitability, the impact, attacker’s capability, masking origin location, and intent. In this Dissertation, we model and analyze scenarios wherein an attacker employs multiple strategies to choose stepping stones. The problem is modeled as an Adjacency Quadratic Shortest Path using dynamic vulnerability graphs with multi-agent dynamic system approach. With this approach, the shortest stepping stone path with maximum node degree and the shortest stepping stone path with maximum impact are modeled and analyzed. Because embedded controllers are omnipresent in networks, in this Dissertation as a Risk Mitigation Strategy, a cyber-attack tolerant control strategy for embedded controllers is proposed. A dual redundant control architecture that combines two identical controllers that are switched periodically between active and restart modes is proposed. The strategy is addressed to mitigate the impact due to the corruption of the controller software by an adversary. We analyze the impact of the resetting and restarting the controller software and performance of the switching process. The minimum requirements in the control design, for effective mitigation of cyber-attacks to the control software that implies a “fast” switching period is provided. The simulation results demonstrate the effectiveness of the proposed strategy when the time to fully reset and restart the controller is faster than the time taken by an adversary to compromise the controller. The results also provide insights into the stability and safety regions and the factors that determine the effectiveness of the proposed strategy