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Fixed-Parameter Tractable Certified Algorithms for Covering and Dominating in Planar Graphs and Beyond

Authors Benjamin Merlin Bumpus , Bart M. P. Jansen , Jaime Venne

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Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Fixed parameter tractability
Keywords
  • fixed-parameter tractability
  • certified algorithms

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Abstract

For a positive real γ ≥ 1, a γ-certified algorithm for a vertex-weighted graph optimization problem is an algorithm that, given a weighted graph (G,w), outputs a re-weighting of the graph obtained by scaling each weight individually with a factor between 1 and γ, along with a solution which is optimal for the perturbed weight function. Here we provide (1+ε)-certified algorithms for Dominating Set and H-Subgraph-Free-Deletion which, for any ε > 0, run in time f(1/ε)⋅n^𝒪(1) on minor-closed classes of graphs of bounded local tree-width with polynomially-bounded weights. We obtain our algorithms as corollaries of a more general result establishing FPT-time certified algorithms for problems admitting, at an intuitive level, certain "local solution-improvement properties". These results improve - in terms of generality, running time and parameter dependence - on Angelidakis, Awasthi, Blum, Chatziafratis and Dan’s XP-time (1+ε)-certified algorithm for Independent Set on planar graphs (ESA2019). Furthermore, our methods are also conceptually simpler: our algorithm is based on elementary local re-optimizations inspired by Baker’s technique, as opposed to the heavy machinery of the Sherali-Adams hierarchy required in previous work.

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Benjamin Merlin Bumpus, Bart M. P. Jansen, and Jaime Venne. Fixed-Parameter Tractable Certified Algorithms for Covering and Dominating in Planar Graphs and Beyond. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SWAT.2024.19

Author Details

Benjamin Merlin Bumpus
  • University of Florida, Gainesville, FL, USA
Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Jaime Venne
  • Eindhoven University of Technology, The Netherlands

Funding

Authors Bumpus and Jansen received funding by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 803421, ReduceSearch).

Acknowledgements

We are grateful to the anonymous referees for pointing out Theorem 2.5 and making suggestions that improved the presentation of the paper.

References

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