Shortest Path Network Interdiction Research Articles

Shortest path network interdiction with asymmetric uncertainty

AbstractThis paper considers an extension of the shortest path network interdiction problem that incorporates robustness to account for parameter uncertainty. The shortest path interdiction problem is a game of two players with conflicting agendas and capabilities: an evader, who traverses the arcs of a network from a source node to a sink node using a path of shortest length, and an interdictor, who maximizes the length of the evader's shortest path by interdicting arcs on the network. It is usually assumed that the parameters defining the network are known exactly by both players. We consider the situation where the evader assumes the nominal parameter values while the interdictor uses robust optimization techniques to account for parameter uncertainty or sensor degradation. We formulate this problem as a nonlinear mixed‐integer semi‐infinite bilevel program and show that it can be converted into a mixed‐integer linear program with a second order cone constraint. We use random geometric networks and transportation networks to perform computational studies and demonstrate the unique decision strategies that our variant produces. Solving the shortest path interdiction problem with asymmetric uncertainty protects the interdictor from investing in a strategy that hinges on key interdictions performing as promised. It also provides an alternate strategy that mitigates the risk of these worst‐case possibilities.

Solution algorithms for shortest path network interdiction with symmetric and asymmetric information

We study symmetric and asymmetric models of shortest path network interdiction and propose an algorithm to solve the problem in each case. We consider the situation where the arcs in a network may be interdicted with different levels of intensity, proportional to the budget spent on interdiction. It is assumed in the asymmetric model that the evader’s perception of the interdiction impact is different from that of the interdictor and thus, each player assumes a different objective function. The first algorithm enhances the efficiency of the decomposition based method for the symmetric case by exploiting the problem structure. The second algorithm is devised for solving the asymmetric model and equipped with a customised supplementary procedure to overcome the issue of trapping in a near-optimal solution. Both symmetric and asymmetric models have been solved to optimality using the proposed algorithms. Furthermore, a duality based approach is employed to reformulate both models and solve the resulting problems using a commercial solver. The proposed algorithms significantly outperform the duality based approach for solving large instances and instances with large interdiction budgets.

Shortest path network interdiction with incomplete information: a robust optimization approach

Shortest path network interdiction with incomplete information: a robust optimization approach

Providing homeland security strategies against interdictions in the city transportation network: A case study in Turkey

This study defines a capacitated multiple-source multiple-sink shortest path problem and introduces its extension, called the capacitated multiple-source multiple-sink shortest path network interdiction problem (CMSSNIP). CMSSNIP examines the actions of attackers who attempt to maximize the total shortest path of network users trying to reach the crime locations for the aid process after causing an incident in certain regions to provide strategic information for the defense systems of the government. In this context, the exact mathematical model is proposed to ensure useful information about safe routes to network users. In this manner, to the best knowledge of authors, the CMSSNIP consisting of multiple-source nodes and multiple-sink nodes and considering capacity-demand relations between security units and crime locations is studied for the first time. Consequently, a set of scenarios is considered based on the levels of the interdiction budget and the number of crime locations through a real case application to show the applicability of the model. Furthermore, computational experiments are performed to evaluate the performance of the model in networks of different sizes. It is realized that the model provides resilient strategies against interdictions in terms of obtaining the safe shortest paths at the operational level within seconds in the real case applications.

Bi-Layer Shortest-Path Network Interdiction Game for Internet of Things

Network security is a crucial challenge facing Internet-of-Things (IoT) systems worldwide, which leads to serious safety alarms and great economic loss. This paper studies the problem of malicious interdicting network exploitation of IoT systems that are modeled as a bi-layer logical–physical network. In this problem, a virtual attack takes place at the logical layer (the layer of Things), while the physical layer (the layer of Internet) provides concrete support for the attack. In the interdiction problem, the attacker attempts to access a target node on the logical layer with minimal communication cost, but the defender can strategically interdict some key edges on the physical layer given a certain budget of interdiction resources. This setting generalizes the classic single-layer shortest-path network interdiction problem, but brings in nonlinear objective functions, which are notoriously challenging to optimize. We reformulate the model and apply Benders decomposition process to solve this problem. A layer-mapping module is introduced to improve the decomposition algorithm and a random-search process is proposed to accelerate the convergence. Extensive numerical experiments demonstrate the computational efficiency of our methods.

The Bi-Objective Shortest Path Network Interdiction Problem: Subgraph Algorithm and Saturation Property

Critical network defense and attack problems, usually modeled as network interdiction, is of great significance. It is of great interest from the interdictor's view to study saturation property of resource allocations. This paper presents a bi-objective shortest path network interdiction model with node interdiction, in which the interdictor seeks to interdict a set of nodes in the network that is Pareto-optimal with respect to two objectives, i.e., maximization of shortest path length and minimization of resources cost. A novel subgraph sequence algorithm is developed to identify the efficient frontier through a sequence of single-objective problems. Novel subgraph decomposition algorithms are proposed to solve single-objective problems. Features and time complexity of those problems and algorithms are analyzed. Moreover, to make a trade-off between the resource investment and the interdiction performance, the definition and the computing method of saturation point are introduced in this paper. Both simulated grid network and real-world transportation network data are used to evaluate the performance of algorithms in numerical experiments. The results show that the subgraph sequence algorithm can achieve the optimal Pareto-frontier of this bi-objective problem effectively.

Robust location of hidden interdictions on a shortest path network

We study a version of the shortest path network interdiction problem in which the follower seeks a path of minimum length on a network and the leader seeks to maximize the follower’s path length by interdicting arcs. We consider placement of interdictions that are not visible to the follower; however, we seek to locate interdictions in a manner that is robust against the possibility that some information about the interdictions becomes known to the follower. We formulate the problem as a bilevel program and derive some properties of the inner problem, which enables solving the problem optimally via a Benders decomposition approach. We derive supervalid inequalities to improve the performance of the algorithm and test the performance of the algorithm on randomly generated, varying-sized grid networks and acyclic networks. We apply our approach to investigate the tradeoffs between conservative (i.e., the follower discovers all interdiction locations) and risky (i.e., the follower discovers no interdiction locations) assumptions regarding the leader’s information advantage.

Online probabilistic goal recognition and its application in dynamic shortest-path local network interdiction

Goal recognition is the task of inferring an agent’s goals given some or all of the agent’s observed actions. However, few research focuses on how to improve the usage effectiveness of knowledge produced by a goal recognition system. In this work, we propose a probabilistic goal recognition approach tailored to a dynamic shortest-path network interdiction problem. Apart from inferring a probabilistic distribution over the possible goals of an agent, our work has another four key novelties: (i) a dynamic shortest-path local network interdiction model that allocates resources locally per step using goal recognition information; (ii) two behavior modeling approaches, including a data-driven learning method based on Inverse Reinforcement Learning as well as a heuristic method taking advantage of the network information, to help solve both the data-intensive and no available data situations; (iii) a heuristic named Subjective Confidence that uses variance in particle system for flexible resource allocation adjustment. The empirical test results show the effectiveness of our goal recognition method, and also verify the practical implications of these methods in solving scalable multi-terminus network interdiction problem.

A Decomposition Approach for Stochastic Shortest-Path Network Interdiction with Goal Threshold

Shortest-path network interdiction, where a defender strategically allocates interdiction resource on the arcs or nodes in a network and an attacker traverses the capacitated network along a shortest s-t path from a source to a terminus, is an important research problem with potential real-world impact. In this paper, based on game-theoretic methodologies, we consider a novel stochastic extension of the shortest-path network interdiction problem with goal threshold, abbreviated as SSPIT. The attacker attempts to minimize the length of the shortest path, while the defender attempts to force it to exceed a specific threshold with the least resource consumption. In our model, threshold constraint is introduced as a trade-off between utility maximization and resource consumption, and stochastic cases with some known probability p of successful interdiction are considered. Existing algorithms do not perform well when dealing with threshold and stochastic constraints. To address the NP-hard problem, SSPIT-D, a decomposition approach based on Benders decomposition, was adopted. To optimize the master problem and subproblem iteration, an efficient dual subgraph interdiction algorithm SSPIT-S and a local research based better-response algorithm SSPIT-DL were designed, adding to the SSPIT-D. Numerical experiments on networks of different sizes and attributes were used to illustrate and validate the decomposition approach. The results showed that the dual subgraph and better-response procedure can significantly improve the efficiency and scalability of the decomposition algorithm. In addition, the improved enhancement algorithms are less sensitive and robust to parameters. Furthermore, the application in a real-world road network demonstrates the scalability of our decomposition approach.

Stochastic network interdiction with incomplete preference

AbstractWe study a class of stochastic network interdiction problems where the defender has incomplete (ambiguous) preferences. Specifically, we focus on the shortest path network interdiction modeled as a Stackelberg game, where the defender (leader) makes an interdiction decision first, and then the attacker (follower) selects a shortest path after the observation of random arc costs and interdiction effects in the network. We assume that the defender's risk preferences over exogenously given probabilities can be summarized by the expected utility theory. Although the exact form of the utility function is ambiguous to the defender, we assume that a set of pairwise gamble comparisons made by the defender is available, which can be used to restrict the shape of the utility function. We present two approaches to tackle this problem. The first approach conducts utility estimation and optimization separately, by first finding the best fit for a piecewise linear concave utility function according to the available data, and then optimizing the expected utility. The second approach integrates utility estimation and optimization, by modeling the utility ambiguity under a robust optimization framework following Armbruster and Delage [B. Armbruster and E. Delage, Manag. Sci., 61 (2015), 111‐128] and Hu and Mehrotra [J. Hu and S. Mehrotra, IIE Trans., 47 (2015), 358‐372]. We conduct extensive computational experiments to evaluate the performances of these approaches.

Stochastic shortest path network interdiction with a case study of Arizona–Mexico border

One of the key challenges in securing the U.S.-Mexico border is the smuggling of illicit goods and humans between Ports-of-Entry (POEs). A confluence of factors advantageous to traffickers including inconsistent levels of fencing, favorable terrain, and expansive knowledge of specific pathways have contributed to the establishment of preferred routes of illicit transit, yet little is known about the strategic interaction between adversaries and defenders between the POEs. To address this challenge, this paper studies a stochastic shortest-path network interdiction problem where the attacker (drug smugglers, illegal immigrants, or terrorists) attempts to minimize the expected shortest traveling time between the source and the destination, while the defender attempts to maximize the attacker’s expected shortest traveling time by allocating sensors to the arcs to detect the attacker with a limited budget. Using a probabilistic detection likelihood, we formulate bi-level max-min mixed-integer problems on a multi-modal licit and illicit transportation network along the Arizona–Mexico border considering single source and single destination, and multiple sources and multiple destinations, respectively. We find that (a) the expected shortest traveling time will increase as the budget/detection probability increase; (b) the expected shortest time by walking is more than 3 times long than by driving; (c) the multiple sources and multiple destinations model which allows the attacker to choose a random source-destination pair leads to a shorter time than the single source and single destination model. A graphical user interface (GUI) is developed to assist decision making and demonstrate the results.

Trilevel shortest path network interdiction with partial fortification

This paper presents a new formulation and solution method for partial fortification and interdiction of a trilevel shortest path problem that is a generalization of existing network interdiction models to a more practical setting. We consider a trilevel problem in which a defense plan is determined at the top level to reinforce some network components against a potential attack. The middle-level problem is solved to predict the most damaging potential attack plan to the hardened network, while the bottom-level problem is used to determine the shortest path in the surviving network. In the proposed model, partial interdiction and fortification are permitted to make the model more general and practical. In addition, the costs of interdiction and fortification are minimized in the attacker’s and defender’s objective functions, and the corresponding budget constraints are relaxed.We have devised a trilevel decomposition algorithm to solve the proposed model in a reasonable time. Furthermore, we transform the trilevel partial fortification problem into a bilevel problem using duality and propose a bilevel decomposition algorithm to solve it. We also study three different formulations for a trilevel shortest path interdiction problem with total fortification and solve these problems using a combination of duality, decomposition and relaxation methods. The numerical results show that the optimal objective values obtained by the proposed models are better than those of the total interdiction models. The solution times of the proposed algorithm applied on a set of test problems are significantly less than those of the existing algorithms. This implies that the budget relaxations in the proposed model improve the total traversing cost in the presence of partial attack and fortification.

An Extended Network Interdiction Problem for Optimal Toll Control

We study an extension of the shortest path network interdiction problem and present a novel real-world application in this area. We consider the problem of determining optimal locations for toll control stations on the arcs of a transportation network. We handle the fact that drivers can avoid control stations on parallel secondary roads. The problem is formulated as a mixed integer program and solved using Benders decomposition. We present experimental results for the application of our models to German motorways.

Geographic characteristics of a network interdiction problem

The protection of critical infrastructure from natural and intentional events is a key component of any national security agenda. Protection schemes need to be readily identifiable and adaptable to complex changing environments. In this paper, we identify strategic geographic characteristics that impact the location of detection resources (e.g. sensors) towards the defense of regional critical infrastructure. Specifically, we seek to estimate the relationship between the results of a variation of the traditional shortest path network interdiction problem and geographical characteristics of the transportation infrastructure and the urban environment. Experiments conducted on three distinct transportation networks of different shapes and granularities (New York City—grid, Houston—radial, Boston—hybrid) underline the importance of geographic characteristics such as the proximity to resource location, attacker entry points as well as network coverage. Insights gained from this work are relevant to policy and decision makers to facilitate the development of analytical and decision-support tools capable of identifying resource allocation strategies. We discuss a heuristic-based framework that prioritizes the selection of detection resources, reflecting the importance of geographic characteristics. The findings underline the importance of geographical characteristics for the allocation of resources in a regional setting.

A Spatial Segmentation Algorithm for Resource Allocation in an Integrated Spatial and Networked Environment

We examine the discrete shortest path network interdiction problem and introduce an algorithm to determine defense resource location points based on network properties. Our proposed method examines how network density, complexity and connectivity can be used to establish improved discrete geographic locations for allocating defense resources. We test performance of the algorithmically-derived locations on the real-world sub-networks of Washington DC, Houston and New York City and compare the resulting discrete shortest path network interdiction results to three grid-based location schemes using computational and spatial analysis techniques. Results demonstrate improved objective performance and a high degree of spatial dissimilarity between the algorithmic and grid-based location schemes. The observed dissimilarity illustrates the importance of strong resource location identification schemes and the propensity for erroneous decision-making under simplified resource location assumptions.

A length-based, multiple-resource formulation for shortest path network interdiction problems in the transportation sector

This paper analyzes a variation of the shortest path network interdiction problem for homeland security scenarios pertaining to attacks on critical infrastructure and key resources that use highways in the transportation sector as conduits for gaining proximity to targets. The model represents a static Stackelberg game and may be formulated as a bi-level mixed integer program with two players: an attacker and a defender. Using highway segments as arcs, a set of predetermined highway entry points and a target set, the attacker seeks the path of maximum non-detection between any entry and target node. The defender impacts the minimum value of this maximum non-detection path through the allocation of a limited number of defense sensors that reduce the non-detection probabilities for arcs that fall within the range of influence of a sensor. Two types of sensors, static and dynamic sensors, are available to the defender and separate influence functions model their respective effects on arc non-detection. A geographic information system is used to collect, store and process network information and sensor influence information stored in a relational database. The results of the problem formulation are analyzed in a case study involving a California highway sub-network. The case study also examines the effects of sensor parameters, budget levels and target sets on the solutions that are obtained.

A constrained binary knapsack approximation for shortest path network interdiction

A modified shortest path network interdiction model is approximated in this work by a constrained binary knapsack which uses aggregated arc maximum flow as the objective function coefficient. In the modified shortest path network interdiction problem, an attacker selects a path of highest non-detection probability on a network with multiple origins and multiple available targets. A defender allocates a limited number of resources within the geographic region of the network to reduce the maximum network non-detection probability between all origin-target pairs by reducing arc non-detection probabilities and where path non-detection probability is modeled as a product of all arc non-detection probabilities on that path. Traditional decomposition methods to solve the shortest path network interdiction problem are sensitive to problem size and network/regional complexity. The goal of this paper is to develop a method for approximating the regional allocation of defense resources that maintains accuracy while reducing both computational effort and the sensitivity of computation time to network/regional properties. Statistical and spatial analysis methods are utilized to verify approximation performance of the knapsack method in two real-world networks.

Role of Spatial Data in the Protection of Critical Infrastructure and Homeland Defense

This paper examines how the use of spatial analysis techniques and geographic information systems may assist in developing a more complete understanding of underlying solutions trends and properties in complex optimization problems. In the model, an attacker with multiple attractive targets and multiple entry points into the region attempts to find a complete network path offering the highest probability of non-detection by a defender who has allocated detection resources to reduce arc non-detection and who wishes to minimize this maximum attacker path (this problem is a direct variant of the shortest path network interdiction problem). This paper introduces spatial analysis techniques to analyze defender resource allocation and attacker path selection in two sample networks (Lancaster-Palmdale, California and Northridge, California). The resulting discussion begins to examine how network diversity and other problem parameters can impact observed defender/attacker solutions and illustrates the importance of this knowledge in the making and evaluating of governmental and public policy decisions.

A bi-objective approach for shortest-path network interdiction

In the literature, solution approaches to the shortest-path network interdiction problem have been developed for optimizing a single figure-of-merit of the network configuration when considering limited amount of resources available to interdict network links. This paper presents a newly developed evolutionary algorithm that allows approximating the optimal Pareto set of network interdiction strategies when considering bi-objective shortest path problems. Thus, the paper considers the concurrent optimization of two objectives: (1) maximization of shortest-path length and (2) minimization of interdiction strategy cost. Also, the paper considers the transformation of the first objective into the minimization of the most reliable path reliability. To solve these multi-objective optimization problems, an evolutionary algorithm has been developed. This algorithm is based on Monte Carlo simulation, to generate potential network interdiction strategies, graph theory to analyze strategies’ shortest path or most reliable path and, an evolutionary search driven by the probability that a link will appear in the optimal Pareto set. Examples for different sizes of networks and network behavior are used throughout the paper to illustrate and validate the approach.

Shortest path network interdiction with asymmetric information

AbstractWe consider an extension of the shortest path network interdiction problem. In this problem an evader attempts to minimize the length of the shortest path between the origin and the destination in a network, while an interdictor attempts to maximize the length of this shortest path by interdicting network arcs using limited resources. We consider the case where there is asymmetric information, i.e., the evader and the interdictor have different levels of information about the network. We formulate this problem as a nonlinear mixed integer program and show that this formulation can be converted to a linear mixed integer program. Computational results demonstrate improvements in the objective function values over the shortest path network interdiction problem with symmetric information. © 2008 Wiley Periodicals, Inc. NETWORKS, 2008