Decremental Sensitivity Oracles for Covering and Packing Minors

Authors Lawqueen Kanesh , Fahad Panolan , M. S. Ramanujan , Peter Strulo



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Author Details

Lawqueen Kanesh
  • Indian Institute of Technology Jodhpur, India
Fahad Panolan
  • School of Computing, University of Leeds, UK
M. S. Ramanujan
  • University of Warwick, UK
Peter Strulo
  • University of Warwick, UK

Acknowledgements

We thank anonymous reviewers for the pointers to [Bruno Courcelle and R. Vanicat, 2003; Wojciech Kazana, 2013].

Cite As Get BibTex

Lawqueen Kanesh, Fahad Panolan, M. S. Ramanujan, and Peter Strulo. Decremental Sensitivity Oracles for Covering and Packing Minors. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 44:1-44:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.STACS.2024.44

Abstract

In this paper, we present the first decremental fixed-parameter sensitivity oracles for a number of basic covering and packing problems on graphs. In particular, we obtain the first decremental sensitivity oracles for Vertex Planarization (delete k vertices to make the graph planar) and Cycle Packing (pack k vertex-disjoint cycles in the given graph). That is, we give a sensitivity oracle that preprocesses the given graph in time f(k,𝓁)n^{{O}(1)} such that, when given a set of 𝓁 edge deletions, the data structure decides in time f(k,𝓁) whether the updated graph is a positive instance of the problem. These results are obtained as a corollary of our central result, which is the first decremental sensitivity oracle for Topological Minor Deletion (cover all topological minors in the input graph that belong to a specified set, using k vertices). 
Though our methodology closely follows the literature, we are able to produce the first explicit bounds on the preprocessing and query times for several problems. We also initiate the study of fixed-parameter sensitivity oracles with so-called structural parameterizations and give sufficient conditions for the existence of fixed-parameter sensitivity oracles where the parameter is just the treewidth of the graph. In contrast, all existing literature on this topic and the aforementioned results in this paper assume a bound on the solution size (a weaker parameter than treewidth for many problems). As corollaries, we obtain decremental sensitivity oracles for well-studied problems such as Vertex Cover and Dominating Set when only the treewidth of the input graph is bounded. A feature of our methodology behind these results is that we are able to obtain query times independent of treewidth.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Sensitivity oracles
  • Data Structures
  • FPT algorithms

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References

  1. Karl Abrahamson and Michael Fellows. Finite automata, bounded treewidth and well-quasiordering. In Graph structure theory (Seattle, WA, 1991), volume 147 of Contemp. Math., pages 539-563, Providence, RI, 1993. Amer. Math. Soc. URL: https://doi.org/10.1090/conm/147/01199.
  2. Isolde Adler, Martin Grohe, and Stephan Kreutzer. Computing excluded minors. In Proceedings of the 19th annual ACM-SIAM symposium on Discrete algorithms (SODA 2008), pages 641-650. SIAM, 2008. URL: http://portal.acm.org/citation.cfm?id=1347082.1347153.
  3. Josh Alman and Dean Hirsch. Parameterized sensitivity oracles and dynamic algorithms using exterior algebras. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 9:1-9:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.9.
  4. 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.
  5. Stefan Arnborg, Jens Lagergren, and Detlef Seese. Easy problems for tree-decomposable graphs. Journal of Algorithms, 12:308-340, 1991. Google Scholar
  6. Davide Bilò, Katrin Casel, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, J. A. Gregor Lagodzinski, Martin Schirneck, and Simon Wietheger. Fixed-parameter sensitivity oracles. In Mark Braverman, editor, 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 31 - February 3, 2022, Berkeley, CA, USA, volume 215 of LIPIcs, pages 23:1-23:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.23.
  7. Václav Blazej, Pratibha Choudhary, Dusan Knop, Jan Matyás Kristan, Ondrej Suchý, and Tomás Valla. Constant factor approximation for tracking paths and fault tolerant feedback vertex set. In Jochen Könemann and Britta Peis, editors, Approximation and Online Algorithms - 19th International Workshop, WAOA 2021, Lisbon, Portugal, September 6-10, 2021, Revised Selected Papers, volume 12982 of Lecture Notes in Computer Science, pages 23-38. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-92702-8_2.
  8. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996. Google Scholar
  9. Hans L. Bodlaender, Fedor V. Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M. Thilikos. (meta) kernelization. J. ACM, 63(5):44:1-44:69, 2016. URL: http://dl.acm.org/citation.cfm?id=2973749, URL: https://doi.org/10.1145/2973749.
  10. Mikolaj Bojanczyk. Separator logic and star-free expressions for graphs. CoRR, abs/2107.13953, 2021. URL: https://arxiv.org/abs/2107.13953.
  11. Richard B. Borie, R. Gary Parker, and Craig A. Tovey. Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica, 7:555-581, 1992. Google Scholar
  12. Cornelius Brand, Holger Dell, and Thore Husfeldt. Extensor-coding. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 151-164. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188902.
  13. Jiehua Chen, Wojciech Czerwinski, Yann Disser, Andreas Emil Feldmann, Danny Hermelin, Wojciech Nadara, Marcin Pilipczuk, Michal Pilipczuk, Manuel Sorge, Bartlomiej Wróblewski, and Anna Zych-Pawlewicz. Efficient fully dynamic elimination forests with applications to detecting long paths and cycles. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 796-809. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.50.
  14. B. Courcelle. The monadic second-order logic of graphs. III. Tree-decompositions, minors and complexity issues. RAIRO Inform. Théor. Appl., 26(3):257-286, 1992. Google Scholar
  15. B. Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In Handbook of graph grammars and computing by graph transformation, Vol. 1, pages 313-400. World Sci. Publ, River Edge, NJ, 1997. URL: https://doi.org/10.1142/9789812384720_0005.
  16. Bruno Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Inform. and Comput., 85:12-75, 1990. Google Scholar
  17. Bruno Courcelle and Joost Engelfriet. Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach. Cambridge University Press, 2012. Google Scholar
  18. Bruno Courcelle and R. Vanicat. Query efficient implementation of graphs of bounded clique-width. Discret. Appl. Math., 131(1):129-150, 2003. URL: https://doi.org/10.1016/S0166-218X(02)00421-3.
  19. 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.
  20. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  21. Camil Demetrescu, Mikkel Thorup, Rezaul Alam Chowdhury, and Vijaya Ramachandran. Oracles for distances avoiding a failed node or link. SIAM J. Comput., 37(5):1299-1318, 2008. Google Scholar
  22. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. Google Scholar
  23. Zdenek Dvorák, Martin Kupec, and Vojtech Tuma. A dynamic data structure for MSO properties in graphs with bounded tree-depth. In Andreas S. Schulz and Dorothea Wagner, editors, Algorithms - ESA 2014 - 22th Annual European Symposium, Wroclaw, Poland, September 8-10, 2014. Proceedings, 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.
  24. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, M. S. Ramanujan, and Saket Saurabh. Solving d-sat via backdoors to small treewidth. In Piotr Indyk, editor, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 630-641. SIAM, 2015. URL: https://doi.org/10.1137/1.9781611973730.43.
  25. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Hitting topological minors is FPT. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 1317-1326. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384318.
  26. Valentin Garnero, Christophe Paul, Ignasi Sau, and Dimitrios M. Thilikos. Explicit linear kernels via dynamic programming. SIAM J. Discret. Math., 29(4):1864-1894, 2015. URL: https://doi.org/10.1137/140968975.
  27. Martin Grohe, Ken-ichi Kawarabayashi, Dániel Marx, and Paul Wollan. Finding topological subgraphs is fixed-parameter tractable. In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 479-488, 2011. URL: https://doi.org/10.1145/1993636.1993700.
  28. Martin Grohe, Ken-ichi Kawarabayashi, and Bruce A. Reed. A simple algorithm for the graph minor decomposition - logic meets structural graph theory. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 414-431. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.30.
  29. Dov Harel and Robert Endre Tarjan. Fast algorithms for finding nearest common ancestors. SIAM J. Comput., 13(2):338-355, 1984. URL: https://doi.org/10.1137/0213024.
  30. Bart M. P. Jansen, Daniel Lokshtanov, and Saket Saurabh. A near-optimal planarization algorithm. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1802-1811, 2014. Google Scholar
  31. Naonori Kakimura and Ken-ichi Kawarabayashi. Fixed-parameter tractability for subset feedback set problems with parity constraints. Theor. Comput. Sci., 576:61-76, 2015. URL: https://doi.org/10.1016/j.tcs.2015.02.004.
  32. Ken-ichi Kawarabayashi. Planarity allowing few error vertices in linear time. In 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, October 25-27, 2009, Atlanta, Georgia, USA, pages 639-648. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/FOCS.2009.45.
  33. Ken-ichi Kawarabayashi and Yusuke Kobayashi. Fixed-parameter tractability for the subset feedback set problem and the s-cycle packing problem. J. Comb. Theory, Ser. B, 102(4):1020-1034, 2012. URL: https://doi.org/10.1016/j.jctb.2011.12.001.
  34. Ken-ichi Kawarabayashi, Yusuke Kobayashi, and Bruce A. Reed. The disjoint paths problem in quadratic time. J. Comb. Theory, Ser. B, 102(2):424-435, 2012. URL: https://doi.org/10.1016/j.jctb.2011.07.004.
  35. Ken-ichi Kawarabayashi and Bruce A. Reed. An (almost) linear time algorithm for odd cyles transversal. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 365-378. SIAM, 2010. URL: https://doi.org/10.1137/1.9781611973075.31.
  36. Ken-ichi Kawarabayashi, Bruce A. Reed, and Paul Wollan. The graph minor algorithm with parity conditions. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 27-36. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.52.
  37. Wojciech Kazana. Query evaluation with constant delay. (L'évaluation de requêtes avec un délai constant). PhD thesis, École normale supérieure de Cachan, Paris, France, 2013. URL: https://tel.archives-ouvertes.fr/tel-00919786.
  38. Jens Lagergren. Upper bounds on the size of obstructions and intertwines. J. Comb. Theory, Ser. B, 73(1):7-40, 1998. URL: https://doi.org/10.1006/jctb.1997.1788.
  39. Christos Levcopoulos, Giri Narasimhan, and Michiel Smid. Efficient algorithms for constructing fault-tolerant geometric spanners. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC '98, pages 186-195, New York, NY, USA, 1998. Association for Computing Machinery. URL: https://doi.org/10.1145/276698.276734.
  40. 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.
  41. William Lochet, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, Roohani Sharma, and Meirav Zehavi. Fault tolerant subgraphs with applications in kernelization. In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference, ITCS 2020, January 12-14, 2020, Seattle, Washington, USA, volume 151 of LIPIcs, pages 47:1-47:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.47.
  42. Konrad Majewski, Michal Pilipczuk, and Marek Sokolowski. Maintaining cmso properties on dynamic structures with bounded feedback vertex number. In Petra Berenbrink, Patricia Bouyer, Anuj Dawar, and Mamadou Moustapha Kanté, editors, 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023, March 7-9, 2023, Hamburg, Germany, volume 254 of LIPIcs, pages 46:1-46:13. Schloss Dagstuhl - Leibniz-Zentrum f"ur Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.STACS.2023.46.
  43. Pranabendu Misra. On fault tolerant feedback vertex set. CoRR, abs/2009.06063, 2020. URL: https://arxiv.org/abs/2009.06063.
  44. Michal Pilipczuk, Nicole Schirrmacher, Sebastian Siebertz, Szymon Torunczyk, and Alexandre Vigny. Algorithms and data structures for first-order logic with connectivity under vertex failures. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 102:1-102:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.102.
  45. Neil Robertson and Paul D. Seymour. Graph minors .XIII. the disjoint paths problem. J. Comb. Theory, Ser. B, 63(1):65-110, 1995. URL: https://doi.org/10.1006/jctb.1995.1006.
  46. Jan van den Brand and Thatchaphol Saranurak. Sensitive distance and reachability oracles for large batch updates. In David Zuckerman, editor, 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 424-435. IEEE Computer Society, 2019. URL: https://doi.org/10.1109/FOCS.2019.00034.
  47. Jan van den Brand and Thatchaphol Saranurak. Sensitive distance and reachability oracles for large batch updates. CoRR, abs/1907.07982, 2019. URL: https://arxiv.org/abs/1907.07982.
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