The Complexity of Blocking All Solutions
Christoph Grüne, Lasse Wulf
TL;DR
This paper studies the minimum cardinality interdiction problem, showing that many interdiction variants are $\Sigma^p_2$-complete when the base problem is NP-hard. It extends the Grüne & Wulf framework to the unit-cost setting via a meta-theorem that provides sufficient conditions for $\Sigma^p_2$-completeness across a wide suite of nominal problems (e.g., satisfiability, dominating set, set cover, knapsack, Hamiltonian path/cycle, TSP, Steiner tree). The results imply that, unless the unlikely conjecture $NP = \Sigma^p_2$ holds, these interdiction problems cannot be modeled by compact integer programs. This work unifies and extends prior $\Sigma^p_2$-completeness results, enabling a single framework to certify hardness for a broad class of interdiction problems.
Abstract
We consider the general problem of blocking all solutions of some given combinatorial problem with only few elements. For example, the problem of destroying all maximum cliques of a given graph by forbidding only few vertices. Problems of this kind are so fundamental that they have been studied under many different names in many different disjoint research communities already since the 90s. Depending on the context, they have been called the interdiction, most vital vertex, most vital edge, blocker, or vertex deletion problem. Despite their apparent popularity, surprisingly little is known about the computational complexity of interdiction problems in the case where the original problem is already NP-complete. In this paper, we fill that gap of knowledge by showing that a large amount of interdiction problems are even harder than NP-hard. Namely, they are complete for the second stage of Stockmeyer's polynomial hierarchy, the complexity class $Σ^p_2$. Such complexity insights are important because they imply that all these problems can not be modelled by a compact integer program (unless the unlikely conjecture NP $= Σ_2^p$ holds). Concretely, we prove $Σ^p_2$-completeness of the following interdiction problems: satisfiability, 3satisfiability, dominating set, set cover, hitting set, feedback vertex set, feedback arc set, uncapacitated facility location, $p$-center, $p$-median, independent set, clique, subset sum, knapsack, Hamiltonian path/cycle (directed/undirected), TSP, $k$ directed vertex disjoint path ($k \geq 2$), Steiner tree. We show that all of these problems share an abstract property which implies that their interdiction counterpart is $Σ_2^p$-complete. Thus, all of these problems are $Σ_2^p$-complete \enquote{for the same reason}. Our result extends a recent framework by Grüne and Wulf.