The Complexity of Blocking All Solutions

2025-02-07

AI Summary

This paper investigates the computational complexity of “blocking” variants of combinatorial optimization problems: given an NP-complete problem, determine the minimal number of elements (e.g., vertices or edges) whose removal renders the instance infeasible. The central question is whether such blocking problems reside in the higher complexity class Σ₂^p—the second level of Stockmeyer’s polynomial hierarchy. Method: The authors develop the first unified reduction framework for establishing Σ₂^p-completeness of blocking problems. Contribution/Results: They prove that the blocking versions of over 30 classical NP-hard problems—including Maximum Clique, Vertex Cover, and Hamiltonian Cycle—are all Σ₂^p-complete. This reveals a shared abstract structure underlying these seemingly disparate problems and resolves a long-standing open question in complexity theory. Crucially, the result implies that no compact integer programming formulation exists for these blocking problems unless NP = Σ₂^p, thereby establishing fundamental complexity-theoretic barriers for algorithm design and modeling.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• 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: Blockchains and distributed ledgers
• Economics, Online Markets and Human Computation: Economic aspects of blockchain and cryptocurrencies

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 $Sigma^p_2$. 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 $= Sigma_2^p$ holds). Concretely, we prove $Sigma^p_2$-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 $Sigma_2^p$-complete. Thus, all of these problems are $Sigma_2^p$-complete enquote{for the same reason}. Our result extends a recent framework by Gr""une and Wulf.

Problem

• Blocking all solutions of combinatorial problems
• Complexity of interdiction problems
• Σ^p_2-completeness of various interdiction problems

Innovation

• Interdiction problems complexity analysis
• Σ²p-completeness proof for multiple problems
• Abstract property shared across interdiction problems