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Sunflowers Meet Sparsity: A Linear-Vertex Kernel for Weighted Clique-Packing on Sparse Graphs

Authors Bart M. P. Jansen , Shivesh K. Roy

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Author Details

Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Shivesh K. Roy
  • Eindhoven University of Technology, The Netherlands

Funding

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 803421, ReduceSearch).

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Bart M. P. Jansen and Shivesh K. Roy. Sunflowers Meet Sparsity: A Linear-Vertex Kernel for Weighted Clique-Packing on Sparse Graphs. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 29:1-29:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.IPEC.2023.29

Abstract

We study the kernelization complexity of the Weighted H-Packing problem on sparse graphs. For a fixed connected graph H, in the Weighted H-Packing problem the input is a graph G, a vertex-weight function w : V(G) → ℕ, and positive integers k, t. The question is whether there exist k vertex-disjoint subgraphs H₁, …, H_k of G such that H_i is isomorphic to H for each i ∈ [k] and the total weight of these k ⋅ |V(H)| vertices is at least t. It is known that the (unweighted) H-Packing problem admits a kernel with 𝒪(k^{|V(H)|-1}) vertices on general graphs, and a linear kernel on planar graphs and graphs of bounded genus. In this work, we focus on case that H is a clique on h ≥ 3 vertices (which captures Triangle Packing) and present a linear-vertex kernel for Weighted K_h-Packing on graphs of bounded expansion, along with a kernel with 𝒪(k^{1+ε}) vertices on nowhere-dense graphs for all ε > 0. To obtain these results, we combine two powerful ingredients in a novel way: the Erdős-Rado Sunflower lemma and the theory of sparsity.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Packing and covering problems
Keywords
  • kernelization
  • weighted problems
  • graph packing
  • sunflower lemma
  • bounded expansion
  • nowhere dense

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