On the completeness of several fortification-interdiction games in the Polynomial Hierarchy

TL;DR

This paper rigorously classifies several fortification-interdiction games within the polynomial hierarchy, establishing their computational complexity and completeness status, which advances understanding of their theoretical foundations.

Contribution

It proves the completeness of multiple well-known fortification problems in specific levels of the polynomial hierarchy, and introduces a general multi-level problem complete for any hierarchy level.

Findings

Several problems are complete for $\\Sigma^p_2$ or $\Sigma^p_3$.

Multi-level problem is complete for any hierarchy level.

Provides a basis for future complexity analyses of protection-interdiction games.

Abstract

Fortification-interdiction games are tri-level 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 Tri-level Interdiction Knapsack Problem with unit fortification and attack weights, the Max-flow Interdiction Problem and Shortest Path Interdiction Problem with Fortification, the Multi-level Critical Node Problem with unit weights, as well as a well-studied electric grid defence planning problem. For all of these problems, we prove their completeness either for the $\Sigma^p_2$ or the $\Sigma^p_3$ class of the polynomial hierarchy. We also prove that the Multi-level Fortification-Interdiction Knapsack Problem with an arbitrary number of protection and interdiction rounds and unit fortification and attack weights 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.