Interdiction Problem Research Articles

The maximum clique interdiction problem

Given a graph G and an interdiction budget k, the Maximum Clique Interdiction Problem asks to find a subset of at most k vertices to remove from G so that the size of the maximum clique in the remaining graph is minimized. This problem has applications in many areas, such as crime detection, prevention of outbreaks of infectious diseases and surveillance of communication networks. We propose an integer linear programming formulation of the problem based on an exponential family of Clique-Interdiction Cuts and we give necessary and sufficient conditions under which these cuts are facet-defining. Our new approach provides a useful tool for analyzing the resilience of (social) networks with respect to clique-interdiction attacks, i.e., the decrease of the size of the maximum clique as a function of an incremental interdiction budget level. On a benchmark set of publicly available instances, including large-scale social networks with up to one hundred thousand vertices and three million edges, we show that most of them can be analyzed and solved to proven optimality within short computing time.

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.

Sequential Interdiction with Incomplete Information and Learning

In “Sequential Interdiction with Incomplete Information and Learning,” J.S. Borrero, O. Prokopyev, and D. Saure study a general class of interdiction problems when the leader has incomplete information regarding the formulation solved by the follower and both interact repeatedly over time. In the setting, the leader learns by observing the follower’s reactions to the leader’s actions. The authors show that strong notions of optimality are not attainable, but a form of weak optimality is attained by the set of proposed policies. These policies are greedy and robust in the sense that the leader reacts “optimally” in each period based on the information at hand and takes a robust approach to handling missing information. The policies are shown to provide a real-time certificate of optimality and to consistently outperform a reasonable benchmark in a series of numerical experiments.

Perfect Matching Interdiction Problem Restricted to a Stable Vertex

Suppose v is the stable vertex of a weighted graph G. The purpose of the perfect matching interdiction problem restricted to the stable vertex v in G is to remove v and a vertex adjacent to v such that the weight of the maximum perfect matching in the resulting graph is minimized. In this paper, we present an approximate solution for this problem. According to the approximate and optimal solutions, a special ratio is introduced for this problem. This ratio is not bounded from above and we obtain it for some special classes of graphs.

A tri-level programming model for attack-resilient control of power grids

The significance of modern power grids is acknowledged every time there is a major threat. This paper proposes the novel approaches to aid power system planner to improve power grid resilience by making appropriate hardening strategies against man-made attack or natural hazards. The vulnerability indices are introduced, which return the most vulnerable component in the system based on a tri-level defender-attacker-operator (DAO) interdiction problem which solves iteratively. The output of DAO is the set of hardening strategies that optimally allocated along the network to mitigate the impact of the worst-case damages. By repeating DAO problem based on the proposed algorithm, the various crafted attack is imposed on the system, and the defender’s behavior demonstrates how an element is vulnerable to threats. The WSCC 9-bus, IEEE 24-bus, and IEEE 118-bus systems are employed to evaluate the model performance. The counter-intuitive results are proven by the proposed robust hardening strategy, which shows how the hardening strategy should be allocated to improve power network resilience against threats.

A Logic-based Decomposition Approach for Multi-Period Network Interdiction Models

This research details the creation of a large-scale optimization approach for solving an application of a multi-period bilevel network interdiction problem (NIP). In this class of multi-period NIP, interdiction activities must be scheduled to minimize the cumulative maximum flow over a finite time horizon. A logic-based decomposition (LBD) approach is proposed that utilizes constraint programming to exploit the scheduling nature of this multi-period NIP. Computational results–comparing solutions obtained using the proposed approach versus traditional mixed-integer programming approach–suggest that the LBD approach is more efficient in finding solutions for medium to large problem instances.

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.

New solution approaches for the maximum-reliability stochastic network interdiction problem

We investigate methods to solve the maximum-reliability stochastic network interdiction problem (SNIP). In this problem, a defender interdicts arcs on a directed graph to minimize an attacker's probability of undetected traversal through the network. The attacker's origin and destination are unknown to the defender and assumed to be random. SNIP can be formulated as a stochastic mixed-integer program via a deterministic equivalent formulation (DEF). As the size of this DEF makes it impractical for solving large instances, current approaches to solving SNIP rely on modifications of Benders decomposition. We present two new approaches to solve SNIP. First, we introduce a new DEF that is significantly more compact than the standard DEF. Second, we propose a new path-based formulation of SNIP. The number of constraints required to define this formulation grows exponentially with the size of the network, but the model can be solved via delayed constraint generation. We present valid inequalities for this path-based formulation which are dependent on the structure of the interdicted arc probabilities. We propose a branch-and-cut (BC) algorithm to solve this new SNIP formulation. Computational results demonstrate that directly solving the more compact SNIP formulation and this BC algorithm both provide an improvement over a state-of-the-art implementation of Benders decomposition for this problem.

Improved formulations for minimum connectivity network interdiction problems

The minimum connectivity interdiction problem seeks to remove at most k nodes from an undirected graph, such that a connectivity measure of the remaining subgraph is minimized. Examples of connectivity measures of a graph include the number of connected pairs of nodes, the size of the largest connected component, and the inverse of the number of connected components. This paper proposes improvements to mixed integer linear programming formulations from the literature. Improvements correspond to the construction of formulations with either a smaller number of constraints, or stronger formulations with increased sparsity of constraint matrices. In addition, two of the discussed problem formulations are demonstrated to provide the same value of linear programming relaxation and hence can be called equivalent. Computational experiments demonstrate that improved formulations allow us to solve some of the considered problem instances up to an order of magnitude times faster than using the recent benchmark.

A bi-objective partial interdiction problem considering different defensive systems with capacity expansion of facilities under imminent attacks

Nowadays, the disordered caused by terrorist attacks is considered as a major treat by governments and international comities. Meanwhile, different types of defensive systems are to develop and to focus on protecting the critical system services in recent years. By this motivation and contrary to previous works, this study firstly proposes a bi-objective, bi-level formulation for the partial interdiction problem considering different defensive systems. In the developed model, defender as a leader tries to optimize the two objectives i.e. different types of defensive systems to protect the facilities as well as the total cost for supply the demands of customers and outsourcing, simultaneously. Besides, the attacker seeks to maximize destroy for the customers' service system. Due to the nature of the problem, the solution approach is based on a nested approach, namely, bi-level metaheuristic. Accordingly, this study not only uses the old and the recent algorithms, but also utilizes two new hybrid algorithms according to the benefits of both groups. The performance of the algorithms is enhanced by Response Surface Method (RSM) to set the proper values for algorithms’ parameters. Finally, the approaches are evaluated by different criteria to identify the performance of algorithms as well as a set of sensitivity analyses are done to approve the efficiency of the proposed model.

Determining the most vital arcs on the shortest path for fire trucks in terrorist actions that will cause fire

In case of fire, the supplying of the water requirements of the fire area is a vital issue. The water requirements must be satisfied as quickly as possible without encountering any obstacles. In the study, once a terrorist attack which will cause fire at certain area (node) is occurred, the situation in which the terrorists want to prevent the fire trucks' transportation to this area via the shortest path is considered. The main logic of the study is determining the risky arc(s) that will interdict and presenting a relatively safety paths for the fire trucks. Terrorists wants to maximize the shortest path of fire trucks depending on limited interdiction budget. In this context, the problem is considered within the framework of the Network Interdiction Problem (NIP), where there are two opposite sides as leader (terrorist) and follower (fire truck). As a result, the bi-level model of the problem is presented first, and the model is applied on a numerical explanatory example.

Dynamic Network Interdiction Problem with Uncertain Data

This paper proposes an uncertain multi-period bi-level network interdiction problem with uncertain arc capacities. It is proved that there exists an equivalence relationship between uncertain multi-period network interdiction problem and the obtained deterministic correspondent. Application of the generalized Benders’ decomposition algorithm is considered as the solution approach to the resulting mixed-integer nonlinear programming problem. Finally, a numerical example is presented to illustrate the model and the algorithm.

Multiple allocation hub interdiction and protection problems: Model formulations and solution approaches

In this paper, we present computationally efficient formulations for the multiple allocation hub interdiction and hub protection problems, which are bilevel and trilevel mixed integer linear programs, respectively. In the hub interdiction problem, the aim is to identify a subset of r critical hubs from an existing set of p hubs that when interdicted results in the maximum post-interdiction cost of routing flows. We present two alternate ways of reducing the bilevel hub interdiction model to a single level optimization problem. The first approach uses the dual formulation of the lower level problem. The second approach exploits the structure of the lower level problem to replace it by a set of closest assignment constraints (CACs). We present alternate sets of CACs, study their dominance relationships, and report their computational performances. Further, we propose refinements to CACs that offer computational advantages of an order-of-magnitude compared to the one existing in the literature. Further, our proposed modifications offer structural advantages for Benders decomposition, which lead to substantial computational savings, particularly for large problem instances. Finally, we study and solve large scale instances of the trilevel hub protection problem exactly by utilizing the ideas developed for the hub interdiction problem.

Interdicting layered physical and information flow networks

ABSTRACTThis article focuses on the problem of interdicting layered networks that involve a physical flow network and an information flow network. There exist dependencies between these networks since components of the physical flow network are only operational should their counterparts in the information flow network receive enough demand. This leads to a network interdiction problem over these layered networks. The objective of the defender is to send the maximum amount of flow through its physical flow network. The objective of the attacker is to interdict components within the layered networks to minimize this maximum flow. For the case where the information supply arcs are uncapacitated, we apply a novel multi-step, dual-based reformulation technique. We apply this reformulation technique to two applications in order to provide policy-driven analysis: law enforcement efforts against illegal drug trafficking networks and cyber vulnerability analysis of infrastructure and supply chain networks. The computational results prove that our reformulation technique outperforms the traditional duality-based reformulation technique by orders of magnitude. This allows us to analyze instances of realistic size.

A polynomial algorithm for a continuous bilevel knapsack problem

In this note, we analyze a bilevel interdiction problem, where the follower’s program is a parametrized continuous knapsack. Based on the structure of the problem and an inverse optimization strategy, we propose for its solution an algorithm with worst-case complexity O(n2).

Shortest Path Network Interdiction With Goal Threshold

Network defense and attack problems can be modeled as network interdictions. In this paper, we consider a new version of the network interdiction problem that optimizes the interdictor’s resource consumption while limiting network capacities (e.g., the shortest path) to a specified threshold. Interdictors would balance the interdiction performance and resource consumption when maximizing the performance costs too much. The most frequently-used technique is to relax the maximization restraint to certain specified threshold; however, the known algorithms provide little help in calculating the threshold restraint. We present a novel model named minimizing the interdiction resource with goal threshold based on the shortest path interdiction problem and propose two basic algorithms to solve it based on Benders decomposition and Lagrange duality, respectively. To consider the situation of complete interdiction and that of a large threshold, a corresponding set-covering algorithm and a Lagrange approximate algorithm are presented. Computational experiments show that decomposition-based algorithms generally perform better on small-world networks while duality-based algorithms perform better on regular grid networks. Analyses on network property sensitivity show that the decomposition-based algorithms are sensitive to the threshold value while the duality-based algorithms become weaker as the interdiction increment grows. Furthermore, we evaluate the performance of these two basic algorithms on a real-world Chicago transportation network. The result illustrates a significant advance; both algorithms can compute an optimal strategy for problems that scale up to real-world sizes.

A novel bilevel model and solution algorithms for multi-period interdiction problem with fortification

Service systems are in significant danger of terrorist attacks aimed at disrupting their critical components. These attacks seek to exterminate vital assets such as transportation networks, services, and supplies. In the present paper, we propose a multi-period planning based on capacity recovery to allocate fortification/interdiction resources in a service system. The problem involves a dynamic Stackelberg game between a defender (leader) and an attacker (follower). The decisions of the defender are the services provided to customers and the fortification resources allocated to facilities in each period as the total demand-weighted distances are minimized. Following this, the attacker allocates interdiction resources to facilities that resulted in the service capacity reduction in each period. In this model, excess fortification/interdiction budgets and capacity in one period can be used in the next period. Moreover, facilities have a predefined capacity to serve the customers with varying demands during the time horizon. To solve this problem, two different types of approaches are implemented and compared. The first method is an exact reformulation algorithm based on the decomposition of the problem into a restricted master problem (RMP) and a slave problem (SP). The second one is a high performance metaheuristic algorithm, i.e., genetic algorithm (GA) developed to overcome the decomposition method’s impracticability on large-scale problem instances. We also compare the results with some novel metaheuristic algorithms such as teaching learning based optimization (TLBO) and dragonfly algorithm (DA). Computational results show the superiority of GA against TLBO and DA.

Hub interdiction problem variants: Models and metaheuristic solution algorithms

Hub location problem (HLP) is one of the strategic problems encountered in designing transportation and telecommunication networks. Regardless of the considered objective in design of hub networks, such as cost minimization or service level maximization, the located hubs can be subject to natural or intentional disruptions after installation. In this paper, we address the multiple allocation p-hub median, p-hub maximal covering, and p-hub center problems under intentional disruptions. In each case, the problem is considered as a Stackelberg game where the leader locates p hubs to optimize his/her objective function, whereas the follower tries to identify and interdict r hubs that their loss would diminish the network performance the most. Bilevel and single level mathematical formulations are presented to model the problem from the leader’s and the follower’s perspectives. Furthermore, efficient Simulated Annealing (SA) heuristics are proposed for solving the problems. Extensive computational experiments show the capability of the proposed SA algorithms to obtain the optimal solutions in short computational times. Some managerial insights are also derived based on the obtained numerical results.

Cutting plane approach for the maximum flow interdiction problem

The maximum flow interdiction is a class of leader–follower optimization problems that seek to identify the set of edges in a network whose interruption minimizes the maximum flow across the network. Particularly, maximum flow interdiction is important in assessing the vulnerability of networks to disruptions. In this paper, the problem is formulated as a bi-level mixed-integer program and an iterative cutting plane algorithm is proposed as a solution methodology. The cutting planes are implemented in a branch-and-cut approach that is computationally effective. Extensive computational results are presented on 336 different instances with varying parameters and with networks of sizes up to 50 nodes, 1200 edge, and 800 commodities. The computational results show that the proposed cutting plane approach has significant computational advantage over the direct solution of the monolithic formulation of the maximum flow interdiction problem for the majority of the tested instances.

A dynamic reformulation heuristic for Generalized Interdiction Problems

We consider a subfamily of mixed-integer linear bilevel problems that we call Generalized Interdiction Problems. This class of problems includes, among others, the widely-studied interdiction problems, i.e., zero-sum Stackelberg games where two players (called the leader and the follower) share a set of items, and the leader can interdict the usage of certain items by the follower. Problems of this type can be modeled as Mixed-Integer Nonlinear Programming problems, whose exact solution can be very hard. In this paper we propose a new heuristic scheme based on a single-level and compact mixed-integer linear programming reformulation of the problem obtained by relaxing the integrality of the follower variables. A distinguished feature of our method is that general-purpose mixed-integer cutting planes for the follower problem are exploited, on the fly, to dynamically improve the reformulation. The resulting heuristic algorithm proved very effective on a large number of test instances, often providing an (almost) optimal solution within very short computing times.