LIPIcs.IPEC.2023.29.pdf
- Filesize: 0.9 MB
- 13 pages
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.
Feedback for Dagstuhl Publishing