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Unified Almost Linear Kernels for Generalized Covering and Packing Problems on Nowhere Dense Classes

Authors Jungho Ahn , Jinha Kim , O-joung Kwon

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Author Details

Jungho Ahn
  • Korea Institute for Advanced Study, Seoul, South Korea
Jinha Kim
  • Department of Mathematics, Chonnam National University, Gwangju, South Korea
O-joung Kwon
  • Department of Mathematics, Hanyang University, Seoul, South Korea
  • Discrete Mathematics Group, Institute for Basic Science, Daejeon, South Korea

Funding

All the authors were supported by the Institute for Basic Science (IBS-R029-C1). Jungho Ahn was also supported by the KIAS Individual Grant (CG095301) at Korea Institute for Advanced Study, and O-joung Kwon was also supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science and ICT (No. NRF-2021K2A9A2A11101617 and RS-2023-00211670).

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Jungho Ahn, Jinha Kim, and O-joung Kwon. Unified Almost Linear Kernels for Generalized Covering and Packing Problems on Nowhere Dense Classes. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.5

Abstract

Let ℱ be a family of graphs, and let p,r be nonnegative integers. For a graph G and an integer k, the (p,r,ℱ)-Covering problem asks whether there is a set D ⊆ V(G) of size at most k such that if the p-th power of G has an induced subgraph isomorphic to a graph in ℱ, then it is at distance at most r from D. The (p,r,ℱ)-Packing problem asks whether G^p has k induced subgraphs H₁,…,H_k such that each H_i is isomorphic to a graph in ℱ, and for i,j ∈ {1,…,k}, the distance between V(H_i) and V(H_j) in G is larger than r. We show that for every fixed nonnegative integers p,r and every fixed nonempty finite family ℱ of connected graphs, (p,r,ℱ)-Covering with p ≤ 2r+1 and (p,r,ℱ)-Packing with p ≤ 2⌊r/2⌋+1 admit almost linear kernels on every nowhere dense class of graphs, parameterized by the solution size k. As corollaries, we prove that Distance-r Vertex Cover, Distance-r Matching, ℱ-Free Vertex Deletion, and Induced-ℱ-Packing for any fixed finite family ℱ of connected graphs admit almost linear kernels on every nowhere dense class of graphs. Our results extend the results for Distance-r Dominating Set by Drange et al. (STACS 2016) and Eickmeyer et al. (ICALP 2017), and for Distance-r Independent Set by Pilipczuk and Siebertz (EJC 2021).

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • kernelization
  • independent set
  • dominating set
  • covering
  • packing

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