Parameterized Critical Node Cut Revisited

TL;DR

This paper thoroughly analyzes the parameterized complexity of Critical Node Cut, revealing new hardness results, identifying parameters that enable fixed-parameter algorithms, and developing an approximation scheme, thereby advancing understanding of graph sparsification problems.

Contribution

The paper provides a comprehensive complexity landscape for Critical Node Cut, including new hardness results, fixed-parameter algorithms for specific parameters, and an FPT approximation scheme.

Findings

W[1]-hardness for combined parameters including feedback edge set, max degree, and pathwidth.

Fixed-parameter tractability for max-leaf number, vertex integrity, and modular-width.

An FPT approximation scheme for treewidth-based solutions.

Abstract

We study how to sparsify connectivity in graphs under a tight deletion budget. Given a graph $G$ and integers $k,x \ge 0$, Critical Node Cut (CNC) asks whether we can delete at most $k$ vertices so that the number of remaining unordered pairs of connected vertices is at most $x$. CNC generalizes Vertex Cover (the case $x=0$) and models tasks in network design, epidemiology, and social network analysis. We comprehensively map the structural parameterized complexity landscape for Critical Node Cut. First, we prove W[1]-hardness for the combined parameter $k + \mathrm{fes} + \Delta + \mathrm{pw}$, where $\mathrm{fes}$ is the feedback edge set number, $\Delta$ the maximum degree, and $\mathrm{pw}$ the pathwidth of the input graph respectively. This significantly improves over the known W[1]-hardness for $k+\mathrm{tw}$, where $\mathrm{tw}$ denotes the treewidth, and is tight in that tree-depth together with maximum degree trivially yields FPT. Second, we give new positive results. Specifically, we identify three structural parameters--max-leaf number, vertex integrity, and modular-width--that render the problem fixed-parameter tractable, and develop a polynomial-time algorithm for graphs of constant clique-width. Third, 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 CNC admits no polynomial kernel when parameterized by vertex cover number, unless standard assumptions fail. Together, these results substantially sharpen the known complexity landscape for CNC.