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Kernelization in Almost Linear Time for Clustering into Bounded Vertex Cover Components

Authors Sriram Bhyravarapu , Pritesh Kumar , Madhumita Kundu , Shivesh K. Roy, Sahiba, Saket Saurabh

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Graph algorithms
Keywords
  • Parameterized complexity
  • Polynomial Kernels
  • Vertex Cover
  • Finite Forbidden Characterization

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Abstract

Motivated by the growing interest in graph clustering and the framework proposed during the Dagstuhl Seminar 23331, we consider a natural specialization of this general approach (as also suggested during the seminar). The seminar introduced a broad perspective on clustering, where the goal is to partition a graph into connected components (or "clusters") that satisfy simple structural integrity constraints - not necessarily limited to cliques.
In our work, we focus on the case where each cluster is required to have bounded vertex cover number. Specifically, a connected component C satisfies this condition if there exists a set S ⊆ V(C) with |S| ≤ d such that C - S is an independent set. We study this within the framework of the {Vertex Deletion to d-Vertex Cover Components} ({Vertex Deletion to d-VCC}) problem: given a graph G and an integer k, the task is to determine whether there exists a vertex set S ⊆ V(G) of size at most k such that every connected component of G - S has vertex cover number at most d. We also examine the edge-deletion variant, {Edge Deletion to d-Vertex Cover Components} ({Edge Deletion to d-VCC}), where the goal is to delete at most k edges so that each connected component of the resulting graph has vertex cover number at most d. We obtain following results.  
1) {Vertex Deletion to d-VCC} admits a kernel with {𝒪}(d⁶k³) vertices and {𝒪}(d⁹k⁴) edges. 
2) {Edge Deletion to d-VCC}, admits a kernel with {𝒪}(d⁴k) vertices and {𝒪}(d⁵k) edges.  Both of our kernelization algorithms run in time 𝒪(1.253^d ⋅ (kd)^{𝒪(1)} ⋅ n log n). It is important to note that, unless the Exponential Time Hypothesis (ETH) fails, the dependence on d cannot be improved to 2^{o(d)}, as the case k = 0 reduces to solving the classical Vertex Cover problem, which is known to require 2^{Ω(d)} time under ETH.
A key ingredient in our kernelization algorithms is a structural result about the hereditary graph class 𝒢_d, consisting of graphs in which every connected component has vertex cover number at most d. We show that 𝒢_d admits a finite obstruction set (with respect to the induced subgraph relation) of size 2^{𝒪(d²)}, where each obstruction graph has at most 3d + 2 vertices. This combinatorial result may be of independent interest.

Cite As Get BibTex

Sriram Bhyravarapu, Pritesh Kumar, Madhumita Kundu, Shivesh K. Roy, Sahiba, and Saket Saurabh. Kernelization in Almost Linear Time for Clustering into Bounded Vertex Cover Components. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.20

Author Details

Sriram Bhyravarapu
  • Indian Institute of Technology Guwahati, India
Pritesh Kumar
  • The Institute of Mathematical Sciences, Chennai, India
Madhumita Kundu
  • University of Bergen, Norway
Shivesh K. Roy
  • The Institute of Mathematical Sciences, Chennai, India
Sahiba
  • Indian Institute of Technology Delhi, New Delhi, India
Saket Saurabh
  • The Institute of Mathematical Sciences, Chennai, India

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