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 ${\mathrm{\Sigma }}_2^p$ or the ${\mathrm{\Sigma }}_3^p$ 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.