Parameterized Capacitated Vertex Cover Revisited

2026-04-20

AI Summary

This study investigates the exact computational complexity of the capacitated vertex cover problem under various structural parameters. Assuming the Exponential Time Hypothesis (ETH), we establish that the known algorithms parameterized by the natural parameter $k$ and treewidth $tw$ are essentially optimal, as no algorithms running in time $k^{o(k)}n^{O(1)}$ or $n^{o(tw)}$ can exist. For vertex integrity $vi$, we present a new algorithm with running time $vi^{O(vi^2)}n^{O(1)}$, which nearly matches its conditional lower bound. Furthermore, we show that the problem remains NP-hard even on graphs of clique-width six, and we demonstrate inherent theoretical barriers to achieving further improvements when parameterized by the vertex cover number $vc$.

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Capacitated Vertex Cover is the hard-capacitated variant of Vertex Cover: given a graph, a capacity for every vertex, and an integer $k$, the task is to select at most $k$ vertices that cover all edges and assign each edge to one of its chosen endpoints so that no chosen vertex receives more incident edges than its capacity. This problem is a classical benchmark in parameterized complexity, as it was among the first natural problems shown to be W[1]-hard when parameterized by treewidth. We revisit its exact complexity from a fine-grained parameterized perspective and obtain a much sharper picture for several standard parameters. For the natural parameter $k$, we prove under the Exponential Time Hypothesis (ETH) that no algorithm with running time $k^{o(k)} n^{\mathcal{O}(1)}$ exists. In particular, this shows that the known algorithms with running time $k^{\mathcal{O}(\mathrm{tw})} n^{\mathcal{O}(1)}$ are essentially optimal. We then turn to more general structural parameters. For vertex cover number $\mathrm{vc}$, we give evidence against a $2^{\mathcal{O}(\mathrm{vc}^{2-\varepsilon})} n^{\mathcal{O}(1)}$ algorithm, as such an improvement would imply corresponding progress for a broader class of integer-programming-type problems. We complement this barrier with a nearly matching upper bound for vertex integrity $\mathrm{vi}$, improving the previously known double-exponential dependence to an algorithm with running time $\mathrm{vi}^{\mathcal{O}(\mathrm{vi}^{2})} n^{\mathcal{O}(1)}$ using $N$-fold integer programming. For treewidth, we show that the standard dynamic programming algorithm with running time $n^{\mathcal{O}(\mathrm{tw})}$ is essentially optimal under the ETH, even if one parameterizes by tree-depth. Turning to clique-width, we prove that Capacitated Vertex Cover remains NP-hard already on graphs of linear clique-width $6$...

Problem

• Capacitated Vertex Cover
• parameterized complexity
• Exponential Time Hypothesis
• treewidth
• clique-width

Innovation

• Capacitated Vertex Cover
• parameterized complexity
• Exponential Time Hypothesis
• N-fold integer programming
• vertex integrity