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Decremental Sensitivity Oracles for Covering and Packing Minors

Authors Lawqueen Kanesh , Fahad Panolan , M. S. Ramanujan , Peter Strulo

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Author Details

Lawqueen Kanesh
  • Indian Institute of Technology Jodhpur, India
Fahad Panolan
  • School of Computing, University of Leeds, UK
M. S. Ramanujan
  • University of Warwick, UK
Peter Strulo
  • University of Warwick, UK

Funding

  • Ramanujan, M. S.: Supported by the Engineering and Physical Sciences Research Council (grant numbers EP/V007793/1 and EP/V044621/1).

Acknowledgements

We thank anonymous reviewers for the pointers to [Bruno Courcelle and R. Vanicat, 2003; Wojciech Kazana, 2013].

Cite AsGet BibTex

Lawqueen Kanesh, Fahad Panolan, M. S. Ramanujan, and Peter Strulo. Decremental Sensitivity Oracles for Covering and Packing Minors. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 44:1-44:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.STACS.2024.44

Abstract

In this paper, we present the first decremental fixed-parameter sensitivity oracles for a number of basic covering and packing problems on graphs. In particular, we obtain the first decremental sensitivity oracles for Vertex Planarization (delete k vertices to make the graph planar) and Cycle Packing (pack k vertex-disjoint cycles in the given graph). That is, we give a sensitivity oracle that preprocesses the given graph in time f(k,𝓁)n^{{O}(1)} such that, when given a set of 𝓁 edge deletions, the data structure decides in time f(k,𝓁) whether the updated graph is a positive instance of the problem. These results are obtained as a corollary of our central result, which is the first decremental sensitivity oracle for Topological Minor Deletion (cover all topological minors in the input graph that belong to a specified set, using k vertices). Though our methodology closely follows the literature, we are able to produce the first explicit bounds on the preprocessing and query times for several problems. We also initiate the study of fixed-parameter sensitivity oracles with so-called structural parameterizations and give sufficient conditions for the existence of fixed-parameter sensitivity oracles where the parameter is just the treewidth of the graph. In contrast, all existing literature on this topic and the aforementioned results in this paper assume a bound on the solution size (a weaker parameter than treewidth for many problems). As corollaries, we obtain decremental sensitivity oracles for well-studied problems such as Vertex Cover and Dominating Set when only the treewidth of the input graph is bounded. A feature of our methodology behind these results is that we are able to obtain query times independent of treewidth.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Sensitivity oracles
  • Data Structures
  • FPT algorithms

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