On the Completeness of Several Fortification-Interdiction Games in the Polynomial Hierarchy

Fortification-interdiction games are trilevel adversarial games where two opponents act in succession to protect, disrupt, and simply use an infrastructure for a specific purpose. Many such games have been formulated and tackled in the literature through specific algorithmic methods; however, very few investigations exist on the completeness of such fortification problems in order to locate them rigorously in the polynomial hierarchy. We clarify the completeness status of several well-known fortification problems, such as the trilevel interdiction knapsack problem with unit fortification and attack costs, the max-flow interdiction problem and shortest path interdiction problem with fortification, the multilevel critical node problem with unit weights, and a well-studied electric grid defence planning problem. For all of these problems, we prove their completeness either for the Sigma p2 or the Sigma p3 class of the polynomial hierarchy. We also prove that the multilevel fortification-interdiction knapsack problem with an arbitrary number of protection and interdiction rounds and unit fortification and attack costs is complete for any level of the polynomial hierarchy, therefore providing a useful basis for further attempts at proving the completeness of protection-interdiction games at any level of said hierarchy.