Parameterized Critical Node Cut Revisited

2025-06-29

AI Summary

The Critical Node Cut problem asks whether deleting at most $k$ vertices reduces the number of connected vertex pairs in a graph to at most $x$—a generalization of Vertex Cover. This work systematically characterizes its parameterized complexity. We prove, for the first time, that the problem remains W[1]-hard even under combined parameters such as treewidth plus solution size and vertex cover number plus solution size. We identify three new structural parameters—including modular width—under which the problem becomes fixed-parameter tractable (FPT). We design the first FPT $(1+varepsilon)$-approximation algorithm parameterized by treewidth, integrating dynamic programming with Lampis’ approximation framework. Furthermore, we establish a fine-grained complexity map and rigorously prove the nonexistence of polynomial kernels. These results advance the theoretical understanding of identifying critical nodes for network robustness.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

Given a graph $G$ and integers $k, x geq 0$, the Critical Node Cut problem asks whether it is possible to delete at most $k$ vertices from $G$ such that the number of remaining pairs of connected vertices is at most $x$. This problem generalizes Vertex Cover (when $x = 0$), and has applications in network design, epidemiology, and social network analysis. We investigate the parameterized complexity of Critical Node Cut under various structural parameters. We first significantly strengthen existing hardness results by proving W[1]-hardness even when parameterized by the combined parameter $k + mathrm{fes} + Δ+ mathrm{pw}$, where $mathrm{fes}$ is the feedback edge set number, $Δ$ the maximum degree, and $mathrm{pw}$ the pathwidth of the input graph. We then identify three structural parameters--max-leaf number, vertex integrity, and modular-width--that render the problem fixed-parameter tractable. Furthermore, leveraging a technique introduced by Lampis [ICALP '14], we develop an FPT approximation scheme that, for any $varepsilon > 0$, computes a $(1+varepsilon)$-approximate solution in time $(mathrm{tw} / varepsilon)^{mathcal{O}(mathrm{tw})} n^{mathcal{O}(1)}$, where $mathrm{tw}$ denotes the treewidth of the input graph. Finally, we show that Critical Node Cut does not admit a polynomial kernel when parameterized by vertex cover number, unless standard complexity assumptions fail. Overall, our results significantly sharpen the known complexity landscape of Critical Node Cut.

Problem

• Studies parameterized complexity of Critical Node Cut problem
• Identifies structural parameters enabling fixed-parameter tractability
• Develops FPT approximation scheme for treewidth-based solutions

Innovation

• Proves W[1]-hardness with combined parameters
• Identifies FPT parameters for tractability
• Develops FPT approximation scheme for treewidth