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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

  1. H. Angelidakis, P. Awasthi, A. Blum, V. Chatziafratis, and C. Dan. Bilu-linial stability, certified algorithms and the independent set problem. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, volume 144 of LIPIcs, pages 7:1-7:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.7.
  2. B. S. Baker. Approximation algorithms for NP-complete problems on planar graphs. J. ACM, 41(1):153-180, January 1994. URL: https://doi.org/10.1145/174644.174650.
  3. Y. Bilu and N. Linial. Are stable instances easy? Comb. Probab. Comput., 21(5):643-660, September 2012. URL: https://doi.org/10.1017/S0963548312000193.
  4. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  5. Marek Cygan, Dániel Marx, Marcin Pilipczuk, and Michał Pilipczuk. Hitting forbidden subgraphs in graphs of bounded treewidth. Information and Computation, 256:62-82, 2017. URL: https://doi.org/10.1016/j.ic.2017.04.009.
  6. M. Delorme, S. García, J. Gondzio, J. Kalcsics, D. Manlove, and W. Pettersson. New algorithms for hierarchical optimisation in kidney exchange programmes. Technical report ERGO 20-005, Edinburgh Research Group in Optimization, 2020. URL: https://optimization-online.org/2020/10/8058/.
  7. Erik D. Demaine and Mohammad Taghi Hajiaghayi. Equivalence of local treewidth and linear local treewidth and its algorithmic applications. In J. Ian Munro, editor, Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, pages 840-849. SIAM, 2004. URL: http://dl.acm.org/citation.cfm?id=982792.982919.
  8. R. Diestel. Graph theory. Springer, 2010. ISBN:9783642142789. Google Scholar
  9. David Eppstein. Diameter and treewidth in minor-closed graph families. Algorithmica, 27(3):275-291, 2000. URL: https://doi.org/10.1007/S004530010020.
  10. Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Bidimensionality and EPTAS. In Dana Randall, editor, Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, pages 748-759. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.59.
  11. M. Grohe. Local tree-width, excluded minors, and approximation algorithms. Combinatorica, 23:613-632, 2000. URL: https://doi.org/10.1007/s00493-003-0037-9.
  12. T. Hazan, G. Papandreou, and D. Tarlow. Bilu-Linial Stability, pages 375-400. The MIT Press, 2016. Google Scholar
  13. Konstantin Makarychev and Yury Makarychev. Perturbation resilience. In Tim Roughgarden, editor, Beyond the Worst-Case Analysis of Algorithms, pages 95-119. Cambridge University Press, 2020. URL: https://doi.org/10.1017/9781108637435.008.
  14. Konstantin Makarychev, Yury Makarychev, and Aravindan Vijayaraghavan. Bilu-linial stable instances of max cut and minimum multiway cut. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, pages 890-906. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.67.
  15. David Manlove. Algorithmics of matching under preferences, volume 2. World Scientific, 2013. Google Scholar
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