Dynamic Complexity Meets Parameterised Algorithms

Authors Jonas Schmidt, Thomas Schwentick, Nils Vortmeier, Thomas Zeume, Ioannis Kokkinis



PDF
Thumbnail PDF

File

LIPIcs.CSL.2020.36.pdf
  • Filesize: 0.59 MB
  • 17 pages

Document Identifiers

Author Details

Jonas Schmidt
  • TU Dortmund University, Dortmund, Germany
Thomas Schwentick
  • TU Dortmund University, Dortmund, Germany
Nils Vortmeier
  • TU Dortmund University, Dortmund, Germany
Thomas Zeume
  • TU Dortmund University, Dortmund, Germany
Ioannis Kokkinis
  • TU Dortmund University, Dortmund, Germany

Acknowledgements

We are grateful to Till Tantau for some valuable discussions.

Cite AsGet BibTex

Jonas Schmidt, Thomas Schwentick, Nils Vortmeier, Thomas Zeume, and Ioannis Kokkinis. Dynamic Complexity Meets Parameterised Algorithms. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 36:1-36:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.CSL.2020.36

Abstract

Dynamic Complexity studies the maintainability of queries with logical formulas in a setting where the underlying structure or database changes over time. Most often, these formulas are from first-order logic, giving rise to the dynamic complexity class DynFO. This paper investigates extensions of DynFO in the spirit of parameterised algorithms. In this setting structures come with a parameter k and the extensions allow additional "space" of size f(k) (in the form of an additional structure of this size) or additional time f(k) (in the form of iterations of formulas) or both. The resulting classes are compared with their non-dynamic counterparts and other classes. The main part of the paper explores the applicability of methods for parameterised algorithms to this setting through case studies for various well-known parameterised problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Logic and databases
  • Theory of computation → Complexity theory and logic
Keywords
  • Dynamic complexity
  • parameterised complexity

Metrics

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

References

  1. Josh Alman, Matthias Mnich, and Virginia Vassilevska Williams. Dynamic Parameterized Problems and Algorithms. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, volume 80 of LIPIcs, pages 41:1-41:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.41.
  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. Max Bannach, Christoph Stockhusen, and Till Tantau. Fast Parallel Fixed-Parameter Algorithms via Color Coding. In 10th International Symposium on Parameterized and Exact Computation, IPEC 2015, pages 224-235. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.IPEC.2015.224.
  4. Max Bannach and Till Tantau. Computing Hitting Set Kernels by AC^0-Circuits. In Rolf Niedermeier and Brigitte Vallée, editors, 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, volume 96 of LIPIcs, pages 9:1-9:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.STACS.2018.9.
  5. Max Bannach and Till Tantau. Computing Kernels in Parallel: Lower and Upper Bounds. In Christophe Paul and Michal Pilipczuk, editors, 13th International Symposium on Parameterized and Exact Computation,IPEC 2018, pages 13:1-13:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.IPEC.2018.13.
  6. Max Bannach and Till Tantau. On the Descriptive Complexity of Color Coding. In Rolf Niedermeier and Christophe Paul, editors, 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019, volume 126 of LIPIcs, pages 11:1-11:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.STACS.2019.11.
  7. David A. Mix Barrington, Neil Immerman, and Howard Straubing. On Uniformity within NC¹. J. Comput. Syst. Sci., 41(3):274-306, 1990. URL: https://doi.org/10.1016/0022-0000(90)90022-D.
  8. Hans-Joachim Böckenhauer, Elisabet Burjons, Martin Raszyk, and Peter Rossmanith. Reoptimization of Parameterized Problems, 2018. URL: http://arxiv.org/abs/1809.10578.
  9. Marco Cesati and Miriam Di Ianni. Parameterized Parallel Complexity. In David J. Pritchard and Jeff Reeve, editors, Euro-Par '98 Parallel Processing, 4th International Euro-Par Conference, Proceedings, volume 1470 of Lecture Notes in Computer Science, pages 892-896. Springer, 1998. URL: https://doi.org/10.1007/BFb0057945.
  10. Yijia Chen and Jörg Flum. Some Lower Bounds in Parameterized AC^0. In Piotr Faliszewski, Anca Muscholl, and Rolf Niedermeier, editors, 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, volume 58 of LIPIcs, pages 27:1-27:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.MFCS.2016.27.
  11. Yijia Chen, Jörg Flum, and Xuangui Huang. Slicewise Definability in First-Order Logic with Bounded Quantifier Rank. In Valentin Goranko and Mads Dam, editors, 26th EACSL Annual Conference on Computer Science Logic, CSL 2017, volume 82 of LIPIcs, pages 19:1-19:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.CSL.2017.19.
  12. Samir Datta, Raghav Kulkarni, Anish Mukherjee, Thomas Schwentick, and Thomas Zeume. Reachability Is in DynFO. J. ACM, 65(5):33:1-33:24, 2018. URL: https://doi.org/10.1145/3212685.
  13. Samir Datta, Anish Mukherjee, Thomas Schwentick, Nils Vortmeier, and Thomas Zeume. A Strategy for Dynamic Programs: Start over and Muddle through. Logical Methods in Computer Science, 15(2), 2019. URL: https://doi.org/10.23638/LMCS-15(2:12)2019.
  14. Larry Denenberg, Yuri Gurevich, and Saharon Shelah. Definability by Constant-Depth Polynomial-Size Circuits. Information and Control, 70(2/3):216-240, 1986. URL: https://doi.org/10.1016/S0019-9958(86)80006-7.
  15. Guozhu Dong, Jianwen Su, and Rodney W. Topor. Nonrecursive Incremental Evaluation of Datalog Queries. Ann. Math. Artif. Intell., 14(2-4):187-223, 1995. URL: https://doi.org/10.1007/BF01530820.
  16. Rodney G. Downey, Judith Egan, Michael R Fellows, Frances A Rosamond, and Peter Shaw. Dynamic Dominating set and Turbo-Charging Greedy Heuristics. Tsinghua Science and Technology, 19(4):329-337, 2014. URL: https://doi.org/10.1109/TST.2014.6867515.
  17. Rodney G. Downey and Michael R. Fellows. Fixed-Parameter Tractability and Completeness I: Basic Results. SIAM J. Comput., 24(4):873-921, 1995. URL: https://doi.org/10.1137/S0097539792228228.
  18. Rodney G. Downey and Michael R. Fellows. Parameterized Computational Feasibility. In Feasible mathematics II, pages 219-244. Springer, 1995. URL: https://doi.org/10.1007/978-1-4612-2566-9_7.
  19. Michael Elberfeld, Christoph Stockhusen, and Till Tantau. On the Space and Circuit Complexity of Parameterized Problems: Classes and Completeness. Algorithmica, 71(3):661-701, 2015. URL: https://doi.org/10.1007/s00453-014-9944-y.
  20. Jörg Flum and Martin Grohe. Describing Parameterized Complexity Classes. Inf. Comput., 187(2):291-319, 2003. URL: https://doi.org/10.1016/S0890-5401(03)00161-5.
  21. Merrick L. Furst, James B. Saxe, and Michael Sipser. Parity, Circuits, and the Polynomial-Time Hierarchy. Mathematical Systems Theory, 17(1):13-27, 1984. URL: https://doi.org/10.1007/BF01744431.
  22. Wouter Gelade, Marcel Marquardt, and Thomas Schwentick. The dynamic complexity of formal languages. ACM Trans. Comput. Log., 13(3):19, 2012. URL: https://doi.org/10.1145/2287718.2287719.
  23. Sepp Hartung and Rolf Niedermeier. Incremental List Coloring of Graphs, Parameterized by Conservation. Theor. Comput. Sci., 494:86-98, 2013. URL: https://doi.org/10.1016/j.tcs.2012.12.049.
  24. Neil Immerman. Descriptive complexity. Graduate texts in computer science. Springer, 1999. URL: https://doi.org/10.1007/978-1-4612-0539-5.
  25. Stefan Kratsch, Geevarghese Philip, and Saurabh Ray. Point Line Cover: The Easy Kernel is Essentially Tight. ACM Trans. Algorithms, 12(3):40:1-40:16, 2016. URL: https://doi.org/10.1145/2832912.
  26. Stefan Langerman and Pat Morin. Covering Things with Things. Discrete & Computational Geometry, 33(4):717-729, 2005. URL: https://doi.org/10.1007/s00454-004-1108-4.
  27. Leonid Libkin. Elements of Finite Model Theory. Springer, 2004. URL: https://doi.org/10.1007/978-3-662-07003-1.
  28. Bernard Mans and Luke Mathieson. Incremental Problems in the Parameterized Complexity Setting. Theory Comput. Syst., 60(1):3-19, 2017. URL: https://doi.org/10.1007/s00224-016-9729-6.
  29. Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Number 31 in Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, 2006. URL: https://doi.org/10.1093/acprof:oso/9780198566076.001.0001.
  30. Sushant Patnaik and Neil Immerman. Dyn-FO: A Parallel, Dynamic Complexity Class. J. Comput. Syst. Sci., 55(2):199-209, 1997. URL: https://doi.org/10.1006/jcss.1997.1520.
  31. Bruce A. Reed, Kaleigh Smith, and Adrian Vetta. Finding Odd Cycle Transversals. Oper. Res. Lett., 32(4):299-301, 2004. URL: https://doi.org/10.1016/j.orl.2003.10.009.
  32. Thomas Schwentick and Thomas Zeume. Dynamic Complexity: Recent Updates. SIGLOG News, 3(2):30-52, 2016. URL: https://doi.org/10.1145/2948896.2948899.
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