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

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 $Σ^p_2$ or the $Σ^p_3$ 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.

💡 Research Summary

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This paper provides a rigorous complexity-theoretic classification of a broad family of Defender‑Attacker‑Defender (DAD) or “fortification‑interdiction” games, which are tri‑level adversarial optimization problems. While many such problems have been studied algorithmically, very few works have placed them precisely within the polynomial hierarchy. The authors address this gap by proving Σ₂^p‑ or Σ₃^p‑completeness for several well‑known fortification‑interdiction problems and by showing that a multi‑level Fortification‑Interdiction Knapsack problem is Σ_k^p‑complete for any integer k, thereby establishing a systematic correspondence between the number of decision rounds and the level of the hierarchy.

The paper begins with a concise review of the polynomial hierarchy, Σ_k^p‑complete problems, and the relevance of quantifier alternation to multi‑stage games. The authors then introduce the canonical Σ₂^p‑complete problem B₂∩3‑CNF (alternating quantified 3‑CNF) and the Σ₃^p‑complete quantified Boolean formula (QBF) as sources for reductions.

Key contributions:

1. Bi‑level Interdiction Knapsack (BIKP) with unit attack costs is shown Σ₂^p‑complete. The reduction encodes each variable and clause of a B₂∩3‑CNF instance into items whose weights and profits are represented as decimal digits placed in distinct positional columns. Selecting items corresponds to fixing truth assignments, and the capacity/profit thresholds enforce the quantified condition.

2. Tri‑level Interdiction Knapsack (TIKP) with unit fortification and attack costs is Σ₃^p‑complete, extending the previous construction by adding a defender’s fortification stage that forces a third quantifier.

3. Generalisation to arbitrary rounds: By iteratively adding defender or attacker rounds, the authors construct a family of Multi‑level Fortification‑Interdiction Knapsack problems that are Σ_k^p‑complete for any k≥2. The proof proceeds by induction on the number of rounds, each step inserting a new quantifier block while preserving the digit‑encoding structure.

4. Maximum‑Flow Interdiction with Fortification (MFIPF) and Shortest‑Path Interdiction with Fortification (SPIPF) are both Σ₂^p‑complete under unit costs. Reductions again use B₂∩3‑CNF, mapping logical structure onto network arcs whose protection status determines flow capacity or shortest‑path length.

5. Multi‑level Critical Node Problem (MCNP), a three‑stage node‑vaccination, node‑attack, and node‑quarantine model, is Σ₃^p‑complete even when all weights and profits are unit. The proof reduces from QBF, encoding quantifier alternation into node selection and removal.

6. Tri‑level Electric Power Grid Fortification Problem (TEPGFP) is shown Σ₂^p‑complete, extending the analysis to a realistic infrastructure‑defence setting.

The technical core of all reductions is a “digit‑encoding” technique: integer weights and profits are constructed as sums of powers of ten, each power representing a distinct logical component (variables, clauses, or quantifier blocks). This ensures that comparisons of total weight or profit against capacity or profit thresholds can be performed digit‑wise without carries, thereby faithfully reproducing the logical constraints of the source problem.

Implications:

• Even with the simplifying assumption of unit fortification and attack costs, DAD problems lie strictly above NP; they occupy higher levels of the polynomial hierarchy. Consequently, exact polynomial‑time algorithms are unlikely, and algorithm designers must rely on approximation, parameterised, or heuristic methods.
• The tight correspondence between the number of decision rounds and the hierarchy level provides a clear theoretical guideline: each additional round introduces an extra quantifier alternation, raising the problem’s complexity by one level. This insight can inform the design of practical defence strategies that deliberately limit the number of interactive stages.
• The results unify a disparate set of interdiction and fortification models under a common complexity framework, facilitating future comparative studies and the transfer of hardness results across domains such as transportation, water distribution, power grids, and security networks.

The paper concludes by summarising the established completeness results, discussing their relevance for both theory and practice, and outlining open directions, notably the exploration of parameterised complexity, approximation thresholds, and the impact of non‑unit cost structures on the hierarchy placement of fortification‑interdiction games.