On the power of randomization in network interdiction

Network interdiction can be viewed as a game between two players, an interdictor and a flow player. The flow player wishes to send as much material as possible through a network, while the interdictor attempts to minimize the amount of transported material by removing a certain number of arcs, say Γ arcs. We introduce the randomized network interdiction problem that allows the interdictor to use randomness to select arcs to be removed. We model the problem in two different ways: arc-based and pathbased formulations, depending on whether flows are defined on arcs or paths, respectively. We present insights into the modeling power, complexity, and approximability of both formulations. In particular, we prove that ZNI/ZRNI ≤ Γ + 1, ZNI/Z RNI ≤ Γ + 1, ZRNI/Z RNI ≤ Γ, where ZNI, ZRNI, and Z Path RNI are the optimal values of the network interdiction problem and its randomized versions in arc-based and path-based formulations, respectively. We also show that these bounds are tight. We show that it is NP-hard to compute the values ZRNI and Z RNI for a general Γ, but they are computable in polynomial time when Γ = 1. Further, we provide a (Γ + 1)-approximation for ZNI, a Γ-approximation for ZRNI, and a ( 1 + ⌊Γ/2⌋ · ⌈Γ/2⌉/(Γ + 1) ) -approximation for Z RNI.