Document Open Access Logo

Sampling with a Black Box: Faster Parameterized Approximation Algorithms for Vertex Deletion Problems

Authors Barış Can Esmer , Ariel Kulik

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2025.39.pdf
  • Filesize: 1.01 MB
  • 20 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Parameterized Approximation Algorithms
  • Random Sampling

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

In this paper, we present Sampling with a Black Box, a unified framework for the design of parameterized approximation algorithms for vertex deletion problems (e.g., Vertex Cover, Feedback Vertex Set, etc.). The framework relies on two components:  
- A Sampling Step. A polynomial-time randomized algorithm that, given a graph G, returns a random vertex v such that the optimum of G⧵ {v} is smaller by 1 than the optimum of G, with some prescribed probability q. We show that such algorithms exist for multiple vertex deletion problems. 
- A Black Box algorithm which is either an exact parameterized algorithm, a polynomial-time approximation algorithm, or a parameterized-approximation algorithm.  The framework combines these two components together. The sampling step is applied iteratively to remove vertices from the input graph, and then the solution is extended using the black box algorithm. The process is repeated sufficiently many times so that the target approximation ratio is attained with a constant probability. 
We use the technique to derive parameterized approximation algorithms for several vertex deletion problems, including Feedback Vertex Set, d-Hitting Set and 𝓁-Path Vertex Cover. In particular, for every approximation ratio 1 < β < 2, we attain a parameterized β-approximation for Feedback Vertex Set, which is faster than the parameterized β-approximation of [Jana, Lokshtanov, Mandal, Rai and Saurabh, MFCS 23']. Furthermore, our algorithms are always faster than the algorithms attained using Fidelity Preserving Transformations [Fellows, Kulik, Rosamond, and Shachnai, JCSS 18'].

Cite As Get BibTex

Barış Can Esmer and Ariel Kulik. Sampling with a Black Box: Faster Parameterized Approximation Algorithms for Vertex Deletion Problems. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 39:1-39:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.39

Author Details

Barış Can Esmer
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Ariel Kulik
  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

  • Kulik, Ariel: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 852780-ERC (SUBMODULAR).

References

  1. Fateme Abbasi, Sandip Banerjee, Jarosław Byrka, Parinya Chalermsook, Ameet Gadekar, Kamyar Khodamoradi, Dániel Marx, Roohani Sharma, and Joachim Spoerhase. Parameterized approximation schemes for clustering with general norm objectives. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 1377-1399. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00085.
  2. Vineet Bafna, Piotr Berman, and Toshihiro Fujito. A 2-Approximation Algorithm for the Undirected Feedback Vertex Set Problem. SIAM Journal on Discrete Mathematics, 12(3):289-297, January 1999. URL: https://doi.org/10.1137/S0895480196305124.
  3. Ann Becker, Reuven Bar-Yehuda, and Dan Geiger. Randomized algorithms for the loop cutset problem. Journal of Artificial Intelligence Research, 12(1):219-234, May 2000. URL: https://doi.org/10.1613/JAIR.638.
  4. Rémy Belmonte, Michael Lampis, and Valia Mitsou. Parameterized (approximate) defective coloring. SIAM Journal on Discrete Mathematics, 34(2):1084-1106, 2020. URL: https://doi.org/10.1137/18M1223666.
  5. Arnab Bhattacharyya, Édouard Bonnet, László Egri, Suprovat Ghoshal, Karthik C. S., Bingkai Lin, Pasin Manurangsi, and Dániel Marx. Parameterized intractability of even set and shortest vector problem. J. ACM, 68(3), March 2021. URL: https://doi.org/10.1145/3444942.
  6. Nicolas Bourgeois, Bruno Escoffier, and Vangelis Th Paschos. Approximation of max independent set, min vertex cover and related problems by moderately exponential algorithms. Discrete Applied Mathematics, 159(17):1954-1970, 2011. URL: https://doi.org/10.1016/J.DAM.2011.07.009.
  7. Ljiljana Brankovic and Henning Fernau. Parameterized Approximation Algorithms for Hitting Set. In Roberto Solis-Oba and Giuseppe Persiano, editors, Approximation and Online Algorithms, pages 63-76, Berlin, Heidelberg, 2012. Springer. URL: https://doi.org/10.1007/978-3-642-29116-6_6.
  8. Ljiljana Brankovic and Henning Fernau. A novel parameterised approximation algorithm for minimum vertex cover. Theoretical Computer Science, 511:85-108, 2013. URL: https://doi.org/10.1016/J.TCS.2012.12.003.
  9. Boštjan Brešar, František Kardoš, Ján Katrenič, and Gabriel Semanišin. Minimum k-path vertex cover. Discrete Applied Mathematics, 159(12):1189-1195, July 2011. URL: https://doi.org/10.1016/j.dam.2011.04.008.
  10. Radovan Červený and Ondřej Suchý. Generating Faster Algorithms for d-Path Vertex Cover. In Daniël Paulusma and Bernard Ries, editors, Graph-Theoretic Concepts in Computer Science, pages 157-171, Cham, 2023. Springer Nature Switzerland. URL: https://doi.org/10.1007/978-3-031-43380-1_12.
  11. Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, and Luca Trevisan. From gap-exponential time hypothesis to fixed parameter tractable inapproximability: Clique, dominating set, and more. SIAM Journal on Computing, 49(4):772-810, 2020. URL: https://doi.org/10.1137/18M1166869.
  12. Yijia Chen, Yi Feng, Bundit Laekhanukit, and Yanlin Liu. Simple combinatorial construction of the k^o(1) -lower bound for approximating the parameterized k-clique. arXiv preprint arXiv:2304.07516, 2023. URL: https://doi.org/10.48550/arXiv.2304.07516.
  13. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer International Publishing, Cham, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  14. Ilan Doron-Arad, Ariel Kulik, and Hadas Shachnai. Budgeted Matroid Maximization: a Parameterized Viewpoint. In Proc. IPEC 2023, pages 13:1-13:17, 2023. URL: https://doi.org/10.4230/LIPICS.IPEC.2023.13.
  15. Pavel Dvorák, Andreas Emil Feldmann, Dusan Knop, Tomás Masarík, Tomas Toufar, and Pavel Veselỳ. Parameterized approximation schemes for steiner trees with small number of steiner vertices. In Proc. STACS, 2018. Google Scholar
  16. Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, Proc. ESA, 2022. URL: https://doi.org/10.4230/LIPIcs.ESA.2022.50.
  17. Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Approximate monotone local search for weighted problems. In Neeldhara Misra and Magnus Wahlström, editors, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023), volume 285 of Leibniz International Proceedings in Informatics (Lipics), pages 17:1-17:23, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.IPEC.2023.17.
  18. Can Barış Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Optimally Repurposing Existing Algorithms to Obtain Exponential-Time Approximations. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Proceedings, pages 314-345. Society for Industrial and Applied Mathematics, January 2024. URL: https://doi.org/10.1137/1.9781611977912.13.
  19. Michael R Fellows, Ariel Kulik, Frances Rosamond, and Hadas Shachnai. Parameterized approximation via fidelity preserving transformations. Journal of Computer and System Sciences, 93:30-40, 2018. URL: https://doi.org/10.1016/J.JCSS.2017.11.001.
  20. Fedor V Fomin, Serge Gaspers, Daniel Lokshtanov, and Saket Saurabh. Exact algorithms via monotone local search. Journal of the ACM (JACM), 66(2):1-23, 2019. URL: https://doi.org/10.1145/3284176.
  21. Fabrizio Grandoni, Stefan Kratsch, and Andreas Wiese. Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack. In Proc. ESA, 2019. Google Scholar
  22. Venkatesan Guruswami, Bingkai Lin, Xuandi Ren, Yican Sun, and Kewen Wu. Parameterized inapproximability hypothesis under ETH. In Proc. STOC (to appear), 2024. Google Scholar
  23. Tanmay Inamdar, Daniel Lokshtanov, Saket Saurabh, and Vaishali Surianarayanan. Parameterized Complexity of Fair Bisection: (FPT-Approximation meets Unbreakability). In Proc. ESA, 2023. Google Scholar
  24. Satyabrata Jana, Daniel Lokshtanov, Soumen Mandal, Ashutosh Rai, and Saket Saurabh. Parameterized Approximation Scheme for Feedback Vertex Set. In Jérôme Leroux, Sylvain Lombardy, and David Peleg, editors, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023), volume 272 of Leibniz International Proceedings in Informatics (LIPIcs), pages 56:1-56:15, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.MFCS.2023.56.
  25. CS Karthik and Subhash Khot. Almost polynomial factor inapproximability for parameterized k-clique. In 37th Computational Complexity Conference (CCC 2022). Schloss-Dagstuhl-Leibniz Zentrum für Informatik, 2022. Google Scholar
  26. Ariel Kulik and Hadas Shachnai. Analysis of two-variable recurrence relations with application to parameterized approximations. In Proc. FOCS, pages 762-773, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00076.
  27. Michael Lampis. Parameterized approximation schemes using graph widths. In International Colloquium on Automata, Languages, and Programming, pages 775-786. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_64.
  28. Jason Li and Jesper Nederlof. Detecting Feedback Vertex Sets of Size k in O⋆ (2.7k) Time. ACM Transactions on Algorithms, 18(4):34:1-34:26, October 2022. URL: https://doi.org/10.1145/3504027.
  29. Bingkai Lin, Xuandi Ren, Yican Sun, and Xiuhan Wang. Constant approximating parameterized k-setcover is W[2]-hard. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3305-3316. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH126.
  30. Bingkai Lin, Xuandi Ren, Yican Sun, and Xiuhan Wang. Improved hardness of approximating k-clique under eth. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), 2023. Google Scholar
  31. Daniel Lokshtanov, Saket Saurabh, and Vaishali Surianarayanan. A parameterized approximation scheme for min k-cut. SIAM Journal on Computing, 2022. Google Scholar
  32. Pasin Manurangsi. A Note on Max k-Vertex Cover: Faster FPT-AS, Smaller Approximate Kernel and Improved Approximation. In Proc. SOSA, 2019. Google Scholar
  33. Pasin Manurangsi. Tight running time lower bounds for strong inapproximability of maximum k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In Proc. SODA, pages 62-81. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.5.
  34. Dekel Tsur. Faster algorithm for pathwidth one vertex deletion. Theoretical Computer Science, 921:63-74, June 2022. URL: https://doi.org/10.1016/j.tcs.2022.04.001.
  35. Jianhua Tu and Wenli Zhou. A factor 2 approximation algorithm for the vertex cover P₃ problem. Information Processing Letters, 111(14):683-686, July 2011. URL: https://doi.org/10.1016/j.ipl.2011.04.009.
  36. Michał Włodarczyk. Parameterized inapproximability for steiner orientation by gap amplification. In Proc. ICALP, 2020. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact