Document Open Access Logo

Parameterized Dynamic Data Structure for Split Completion

Authors Konrad Majewski , Michał Pilipczuk , Anna Zych-Pawlewicz

PDF
Thumbnail PDF

File

LIPIcs.ESA.2024.87.pdf
  • Filesize: 1.54 MB
  • 17 pages

Document Identifiers

Author Details

Konrad Majewski
  • Institute of Informatics, University of Warsaw, Poland
Michał Pilipczuk
  • Institute of Informatics, University of Warsaw, Poland
Anna Zych-Pawlewicz
  • Institute of Informatics, University of Warsaw, Poland

Funding

This work is a part of project BOBR that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 948057).

Cite AsGet BibTex

Konrad Majewski, Michał Pilipczuk, and Anna Zych-Pawlewicz. Parameterized Dynamic Data Structure for Split Completion. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 87:1-87:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.87

Abstract

We design a randomized data structure that, for a fully dynamic graph G updated by edge insertions and deletions and integers k, d fixed upon initialization, maintains the answer to the Split Completion problem: whether one can add k edges to G to obtain a split graph. The data structure can be initialized on an edgeless n-vertex graph in time n ⋅ (k d ⋅ log n)^{𝒪(1)}, and the amortized time complexity of an update is 5^k ⋅ (k d ⋅ log n)^{𝒪(1)}. The answer provided by the data structure is correct with probability 1-𝒪(n^{-d}).

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Data structures design and analysis
Keywords
  • parameterized complexity
  • dynamic data structures
  • split graphs

Metrics

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

Related Versions

References

  1. Josh Alman, Matthias Mnich, and Virginia Vassilevska Williams. Dynamic parameterized problems and algorithms. ACM Trans. Algorithms, 16(4):45:1-45:46, 2020. URL: https://doi.org/10.1145/3395037.
  2. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. ACM, 42(4):844-856, 1995. URL: https://doi.org/10.1145/210332.210337.
  3. Ivan Bliznets, Fedor V. Fomin, Marcin Pilipczuk, and Michał Pilipczuk. A subexponential parameterized algorithm for Proper Interval Completion. SIAM J. Discret. Math., 29(4):1961-1987, 2015. URL: https://doi.org/10.1137/140988565.
  4. Ivan Bliznets, Fedor V. Fomin, Marcin Pilipczuk, and Michał Pilipczuk. Subexponential parameterized algorithm for Interval Completion. ACM Trans. Algorithms, 14(3):35:1-35:62, 2018. URL: https://doi.org/10.1145/3186896.
  5. Yixin Cao. Linear recognition of almost interval graphs. In 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, pages 1096-1115. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch77.
  6. Yixin Cao. Unit interval editing is fixed-parameter tractable. Inf. Comput., 253:109-126, 2017. URL: https://doi.org/10.1016/j.ic.2017.01.008.
  7. Yixin Cao and Dániel Marx. Chordal editing is fixed-parameter tractable. Algorithmica, 75(1):118-137, 2016. URL: https://doi.org/10.1007/s00453-015-0014-x.
  8. Jiehua Chen, Wojciech Czerwiński, Yann Disser, Andreas Emil Feldmann, Danny Hermelin, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk, Manuel Sorge, Bartłomiej Wróblewski, and Anna Zych-Pawlewicz. Efficient fully dynamic elimination forests with applications to detecting long paths and cycles. In 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, pages 796-809. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.50.
  9. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  10. Marek Cygan and Marcin Pilipczuk. Split Vertex Deletion meets Vertex Cover: New fixed-parameter and exact exponential-time algorithms. Inf. Process. Lett., 113(5-6):179-182, 2013. URL: https://doi.org/10.1016/j.ipl.2013.01.001.
  11. Pål Grønås Drange, Fedor V. Fomin, Michał Pilipczuk, and Yngve Villanger. Exploring the subexponential complexity of completion problems. ACM Trans. Comput. Theory, 7(4):14:1-14:38, 2015. URL: https://doi.org/10.1145/2799640.
  12. Pål Grønås Drange and Michal Pilipczuk. A polynomial kernel for Trivially Perfect Editing. Algorithmica, 80(12):3481-3524, 2018. URL: https://doi.org/10.1007/s00453-017-0401-6.
  13. Zdenek Dvořák, Martin Kupec, and Vojtech Tůma. A dynamic data structure for MSO properties in graphs with bounded tree-depth. In 22th Annual European Symposium on Algorithms, ESA 2014, volume 8737 of Lecture Notes in Computer Science, pages 334-345. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-44777-2_28.
  14. Zdenek Dvořák and Vojtech Tůma. A dynamic data structure for counting subgraphs in sparse graphs. In 13th International Symposium on Algorithms and Data Structures, WADS 2013, volume 8037 of Lecture Notes in Computer Science, pages 304-315. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-40104-6_27.
  15. Fedor V. Fomin, Saket Saurabh, and Yngve Villanger. A polynomial kernel for Proper Interval Vertex Deletion. SIAM J. Discret. Math., 27(4):1964-1976, 2013. URL: https://doi.org/10.1137/12089051X.
  16. Fedor V. Fomin and Yngve Villanger. Subexponential parameterized algorithm for Minimum Fill-in. SIAM J. Comput., 42(6):2197-2216, 2013. URL: https://doi.org/10.1137/11085390X.
  17. Stéphane Földes and Peter L. Hammer. Split graphs. In Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing, volume XIX of Congressus Numerantium, pages 311-315, 1977. Google Scholar
  18. Esha Ghosh, Sudeshna Kolay, Mrinal Kumar, Pranabendu Misra, Fahad Panolan, Ashutosh Rai, and M. S. Ramanujan. Faster parameterized algorithms for deletion to split graphs. Algorithmica, 71(4):989-1006, 2015. URL: https://doi.org/10.1007/s00453-013-9837-5.
  19. Alejandro Grez, Filip Mazowiecki, Michał Pilipczuk, Gabriele Puppis, and Cristian Riveros. Dynamic data structures for timed automata acceptance. Algorithmica, 84(11):3223-3245, 2022. URL: https://doi.org/10.1007/s00453-022-01025-8.
  20. Peter L. Hammer and Bruno Simeone. The splittance of a graph. Combinatorica, 1:275-284, 1981. Google Scholar
  21. Pinar Heggernes and Federico Mancini. Dynamically maintaining split graphs. Discret. Appl. Math., 157(9):2057-2069, 2009. URL: https://doi.org/10.1016/J.DAM.2008.06.028.
  22. Louis Ibarra. Fully dynamic algorithms for chordal graphs and split graphs. ACM Trans. Algorithms, 4(4):40:1-40:20, 2008. URL: https://doi.org/10.1145/1383369.1383371.
  23. Yoichi Iwata and Keigo Oka. Fast dynamic graph algorithms for parameterized problems. In 14th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2014, volume 8503 of Lecture Notes in Computer Science, pages 241-252. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-08404-6_21.
  24. Bart M. P. Jansen, Daniel Lokshtanov, and Saket Saurabh. A near-optimal planarization algorithm. In 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, pages 1802-1811. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.130.
  25. Yuping Ke, Yixin Cao, Xiating Ouyang, Wenjun Li, and Jianxin Wang. Unit interval vertex deletion: Fewer vertices are relevant. J. Comput. Syst. Sci., 95:109-121, 2018. URL: https://doi.org/10.1016/j.jcss.2018.01.001.
  26. Tuukka Korhonen, Konrad Majewski, Wojciech Nadara, Michał Pilipczuk, and Marek Sokołowski. Dynamic treewidth. CoRR, abs/2304.01744, 2023. To appear in the proceedings of FOCS 2023. URL: https://doi.org/10.48550/arXiv.2304.01744.
  27. John M. Lewis and Mihalis Yannakakis. The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci., 20(2):219-230, 1980. URL: https://doi.org/10.1016/0022-0000(80)90060-4.
  28. Konrad Majewski, Michał Pilipczuk, and Marek Sokołowski. Maintaining CMSO₂ properties on dynamic structures with bounded feedback vertex number. In 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023, volume 254 of LIPIcs, pages 46:1-46:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.STACS.2023.46.
  29. Konrad Majewski, Michal Pilipczuk, and Anna Zych-Pawlewicz. Parameterized dynamic data structure for Split Completion. CoRR, abs/2402.08816, 2024. https://arxiv.org/abs/2402.08816, URL: https://doi.org/10.48550/arXiv.2402.08816.
  30. Dániel Marx. Chordal deletion is fixed-parameter tractable. Algorithmica, 57(4):747-768, 2010. URL: https://doi.org/10.1007/s00453-008-9233-8.
  31. Dániel Marx and Ildikó Schlotter. Obtaining a planar graph by vertex deletion. Algorithmica, 62(3-4):807-822, 2012. URL: https://doi.org/10.1007/s00453-010-9484-z.
  32. Assaf Natanzon, Ron Shamir, and Roded Sharan. Complexity classification of some edge modification problems. Discret. Appl. Math., 113(1):109-128, 2001. URL: https://doi.org/10.1016/S0166-218X(00)00391-7.
  33. Jędrzęj Olkowski, Michał Pilipczuk, Mateusz Rychlicki, Karol Węgrzycki, and Anna Zych-Pawlewicz. Dynamic data structures for parameterized string problems. In 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023, volume 254 of LIPIcs, pages 50:1-50:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.STACS.2023.50.
  34. Pim van 't Hof and Yngve Villanger. Proper interval vertex deletion. Algorithmica, 65(4):845-867, 2013. URL: https://doi.org/10.1007/s00453-012-9661-3.
  35. Yngve Villanger, Pinar Heggernes, Christophe Paul, and Jan Arne Telle. Interval Completion is fixed parameter tractable. SIAM J. Comput., 38(5):2007-2020, 2009. URL: https://doi.org/10.1137/070710913.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact