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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 ,
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19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Scandinavian Symposium and Workshops on Algorithm Theory (SWAT) - License:
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- Publication Date: 2024-05-31
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Acknowledgements
We are grateful to the anonymous referees for pointing out Theorem 2.5 and making suggestions that improved the presentation of the paper.
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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
https://doi.org/10.4230/LIPIcs.SWAT.2024.19
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.
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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References
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