Protection of flows under targeted attacks

Due to the importance of robustness in many real-world optimization problems, the field of robust optimization has gained a lot of attention over the past decade. We concentrate on maximum flow problems and introduce a novel robust optimization model which, compared to known models from the literature, features several advantageous properties: (i) We consider a general class of path-based flow problems which can be used to model a large variety of network routing problems (and other packing problems). (ii) We aim at solutions that are robust against targeted attacks by a potent adversary who may attack any flow path of his choice on any edge of the network. (iii) In contrast to previous robust maximum flow models, for which no efficient algorithms are known, optimal robust flows for the most important basic variants of our model can be found in polynomial time. We also consider generalizations where the flow player can spend a budget to protect the network against the interdictor. Here, we show that the problem can be solved efficiently when the interdic-tion costs are determined by the flow player from scratch. However, the problem becomes hard to approximate when the flow player has to improve an initial protection infrastructure. 1. Introduction. Network flow problems form one of the most important classes of optimization problems with numerous real-world applications, e. g., in production systems, logistics, and communication networks. The increasing dependence of our society on constant availability of such network services motivates the study of new flow models that are robust against unforeseen interferences, link failures, and targeted attacks by external forces. The theory of robust optimization offers various techniques to handle the issue of planning in face of uncertainties and unreliability; see, e. g., [4, 5] for surveys. A general idea is to model uncertainty by a set of possible scenarios Ω that is specified along with the instance of the optimization problem under consideration, where each scenario represents a possible outcome involving failures in the infrastructure, intentional sabotage, or similar complications. With respect to a worst-case analysis, the robust objective value of a feasible solution x is the worst objective value of x among all possible scenarios z ∈ Ω. Robust optimization can thus be interpreted as a two-player game: the first player (" the decision-maker ") chooses a solution x from a set X of feasible solutions to the underlying " nominal " optimization problem. Afterwards, …