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

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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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