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

2024-06-03

arXiv.org

AI Summary

This paper establishes the exact polynomial hierarchy (PH) completeness of multiple classic three-stage defense–interdiction games, including unit-weight three-level interdiction knapsack, max-flow/shortest-path interdiction, multilevel critical node identification, and power-grid defense planning—six problems previously unclassified in PH. Method: We introduce an extensible multilevel knapsack model and develop structured reduction techniques to anchor Σₖ^p-completeness for arbitrary k ≥ 2. Our approach integrates three-stage game modeling, fine-grained reductions under unit-weight constraints, and rigorous PH-completeness analysis. Contribution/Results: We provide the first unified complexity classification framework and generic constructive toolkit for security-critical adversarial decision-making problems. Specifically, we prove that the six canonical problems are complete for either Σ₂^p or Σ₃^p; moreover, our reduction framework achieves Σₖ^p-completeness for all k ≥ 2. These results establish the foundational computational complexity theory of defensive interdiction games.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Adversarial Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Clarify completeness of fortification-interdiction games in polynomial hierarchy
• Prove Σ^p_2 or Σ^p_3 completeness for specific tri-level adversarial problems
• Establish multi-level knapsack problem as basis for hierarchy completeness

Innovation

• Clarifies completeness of fortification-interdiction games
• Proves completeness for Σ²p and Σ³p classes
• Establishes basis for multi-level hierarchy proofs