Stochastic network interdiction games

Thesis (Ph.D.)--Boston UniversityNetwork interdiction problems consist of games between an attacker and an intelligent network, where the attacker seeks to degrade network operations while the network adapts its operations to counteract the effects of the attacker. This problem has received significant attention in recent years due to its relevance to military problems and network security. When the attacker's actions achieve uncertain effects, the resulting problems become stochastic network interdiction problems. In this thesis, we develop new algorithms for the solutions of different classes of stochastic network interdiction problems. We first focus on static network interdiction games where the attacker attacks the network once, which will change the network with certain probability. Then the network will maximize the flow from a given source to its destination. The attacker is seeking a strategy which minimizes the expected maximum flow after the attack. For this problem, we develop a new solution algorithm, based on parsimonious integration of branch and bound techniques with increasingly accurate lower bounds. Our method obtains solutions significantly faster than previous approaches in the literature. In the second part, we study a multi-stage interdiction problem where the attacker can attack the network multiple times, and observe the outcomes of its past attacks before selecting a current attack. For this dynamic interdiction game, we use a model-predictive approach based on a lower bound approximation. We develop a new set of performance bounds, which are integrated into a modified branch and bound procedure that extends the single stage approach to multiple stages. We show that our new algorithm is faster than other available methods with simulated experiments. In the last part, we study the nested information game between an intelligent network and an attacker, where the attacker has partial information about the network state, which refers to the availability of arcs. The attacker does not know the exact state, but has a probability distribution over the possible network states. The attacker makes several attempts to attack the network and observes the flows on the network. These observations will update the attacker's knowledge of the network and will be used in selecting the next attack actions. The defender can either send flow on that arc if it survived, or refrain from using it in order to deceive the attacker. For these problems, we develop a faster algorithm, which decomposes this game into a sequence of subgames and solves them to get the equilibrium strategy for the original game. Numerical results show that our method can handle large problems which other available methods fail to solve

Decomposition Algorithms in Stochastic Integer Programming: Applications and Computations.

In this dissertation we focus on two main topics. Under the first topic, we develop a new framework for stochastic network interdiction problem to address ambiguity in the defender risk preferences. The second topic is dedicated to computational studies of two-stage stochastic integer programs. More specifically, we consider two cases. First, we develop some solution methods for two-stage stochastic integer programs with continuous recourse; second, we study some computational strategies for two-stage stochastic integer programs with integer recourse. We 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, then the attacker (follower) selects a shortest path after the observation of random arc costs and interdiction effects in the network. We take a decision-analytic perspective in addressing probabilistic risk over network parameters, assuming that the defender\u27s 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 historical data on some pairwise comparisons made by the defender is available, which can be used to restrict the shape of the utility function. We use two different 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 \cite{armbruster2015decision} and \cite{Hu}. We conduct extensive computational experiments to evaluate the performances of these approaches on the stochastic shortest path network interdiction problem. In third chapter, we propose partition-based decomposition algorithms for solving two-stage stochastic integer program with continuous recourse. The partition-based decomposition method enhance the classical decomposition methods (such as Benders decomposition) by utilizing the inexact cuts (coarse cuts) induced by a scenario partition. Coarse cut generation can be much less expensive than the standard Benders cuts, when the partition size is relatively small compared to the total number of scenarios. We conduct an extensive computational study to illustrate the advantage of the proposed partition-based decomposition algorithms compared with the state-of-the-art approaches. In chapter four, we concentrate on computational methods for two-stage stochastic integer program with integer recourse. We consider the partition-based relaxation framework integrated with a scenario decomposition algorithm in order to develop strategies which provide a better lower bound on the optimal objective value, within a tight time limit

Optimal Interdiction of Unreactive Markovian Evaders

The interdiction problem arises in a variety of areas including military logistics, infectious disease control, and counter-terrorism. In the typical formulation of network interdiction, the task of the interdictor is to find a set of edges in a weighted network such that the removal of those edges would maximally increase the cost to an evader of traveling on a path through the network. Our work is motivated by cases in which the evader has incomplete information about the network or lacks planning time or computational power, e.g. when authorities set up roadblocks to catch bank robbers, the criminals do not know all the roadblock locations or the best path to use for their escape. We introduce a model of network interdiction in which the motion of one or more evaders is described by Markov processes and the evaders are assumed not to react to interdiction decisions. The interdiction objective is to find an edge set of size B, that maximizes the probability of capturing the evaders. We prove that similar to the standard least-cost formulation for deterministic motion this interdiction problem is also NP-hard. But unlike that problem our interdiction problem is submodular and the optimal solution can be approximated within 1-1/e using a greedy algorithm. Additionally, we exploit submodularity through a priority evaluation strategy that eliminates the linear complexity scaling in the number of network edges and speeds up the solution by orders of magnitude. Taken together the results bring closer the goal of finding realistic solutions to the interdiction problem on global-scale networks.Comment: Accepted at the Sixth International Conference on integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems (CPAIOR 2009