Document Open Access Logo

Lower Bounds for Dynamic Programming on Planar Graphs of Bounded Cutwidth

Authors Bas A. M. van Geffen, Bart M. P. Jansen , Arnoud A. W. M. de Kroon, Rolf Morel

PDF
Thumbnail PDF

File

LIPIcs.IPEC.2018.3.pdf
  • Filesize: 0.66 MB
  • 14 pages

Document Identifiers

Author Details

Bas A. M. van Geffen
  • University of Oxford
Bart M. P. Jansen
  • Eindhoven University of Technology
Arnoud A. W. M. de Kroon
  • University of Oxford
Rolf Morel
  • University of Oxford

Funding

  • Jansen, Bart M. P.: Supported by NWO Gravitation grant "Networks" and NWO Veni grant "Frontiers in Parameterized Preprocessing".

Cite As Get BibTex

Bas A. M. van Geffen, Bart M. P. Jansen, Arnoud A. W. M. de Kroon, and Rolf Morel. Lower Bounds for Dynamic Programming on Planar Graphs of Bounded Cutwidth. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.IPEC.2018.3

Abstract

Many combinatorial problems can be solved in time O^*(c^{tw}) on graphs of treewidth tw, for a problem-specific constant c. In several cases, matching upper and lower bounds on c are known based on the Strong Exponential Time Hypothesis (SETH). In this paper we investigate the complexity of solving problems on graphs of bounded cutwidth, a graph parameter that takes larger values than treewidth. We strengthen earlier treewidth-based lower bounds to show that, assuming SETH, Independent Set cannot be solved in O^*((2-epsilon)^{ctw}) time, and Dominating Set cannot be solved in O^*((3-epsilon)^{ctw}) time. By designing a new crossover gadget, we extend these lower bounds even to planar graphs of bounded cutwidth or treewidth. Hence planarity does not help when solving Independent Set or Dominating Set on graphs of bounded width. This sharply contrasts the fact that in many settings, planarity allows problems to be solved much more efficiently.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • planarization
  • dominating set
  • cutwidth
  • lower bounds
  • strong exponential time hypothesis

Metrics

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

Related Versions

References

  1. Julien Baste and Ignasi Sau. The role of planarity in connectivity problems parameterized by treewidth. Theor. Comput. Sci., 570:1-14, 2015. URL: http://dx.doi.org/10.1016/j.tcs.2014.12.010.
  2. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets Möbius: Fast subset convolution. In David S. Johnson and Uriel Feige, editors, Proc. 39th STOC, pages 67-74. ACM, 2007. URL: http://dx.doi.org/10.1145/1250790.1250801.
  3. Hans L. Bodlaender. A Partial k-Arboretum of Graphs with Bounded Treewidth. Theor. Comput. Sci., 209(1-2):1-45, 1998. URL: http://dx.doi.org/10.1016/S0304-3975(97)00228-4.
  4. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput., 243:86-111, 2015. URL: http://dx.doi.org/10.1016/j.ic.2014.12.008.
  5. Hans L. Bodlaender and Arie M. C. A. Koster. Combinatorial Optimization on Graphs of Bounded Treewidth. Comput. J., 51(3):255-269, 2008. URL: http://dx.doi.org/10.1093/comjnl/bxm037.
  6. Jianer Chen, Iyad A. Kanj, and Weijia Jia. Vertex Cover: Further Observations and Further Improvements. J. Algorithms, 41(2):280-301, 2001. URL: http://dx.doi.org/10.1006/jagm.2001.1186.
  7. Bruno Courcelle. The Monadic Second-Order Logic of Graphs I: Recognizable Sets of Finite Graphs. Inf. Comput., 85(1):12-75, 1990. URL: http://dx.doi.org/10.1016/0890-5401(90)90043-H.
  8. Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Fast hamiltonicity checking via bases of perfect matchings. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Proc. 45th STOC, pages 301-310. ACM, 2013. URL: http://dx.doi.org/10.1145/2488608.2488646.
  9. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time. In Proc. 52nd FOCS, pages 150-159, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.23.
  10. Erik D. Demaine and MohammadTaghi Hajiaghayi. The Bidimensionality Theory and Its Algorithmic Applications. Comput. J., 51(3):292-302, 2008. URL: http://dx.doi.org/10.1093/comjnl/bxm033.
  11. Josep Díaz, Jordi Petit, and Maria J. Serna. A survey of graph layout problems. ACM Comput. Surv., 34(3):313-356, 2002. URL: http://dx.doi.org/10.1145/568522.568523.
  12. David Eppstein. Pathwidth of planarized drawing of K_3,n. TheoryCS StackExchange question, 2016. URL: http://cstheory.stackexchange.com/questions/35974/.
  13. David Eppstein. The Effect of Planarization on Width. In Proc. 25th GD, volume 10692 of LNCS, pages 560-572, 2017. URL: http://dx.doi.org/10.1007/978-3-319-73915-1_43.
  14. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Efficient Computation of Representative Families with Applications in Parameterized and Exact Algorithms. J. ACM, 63(4):29:1-29:60, 2016. URL: http://dx.doi.org/10.1145/2886094.
  15. M.R. Garey, D.S. Johnson, and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1(3):237-267, 1976. URL: http://dx.doi.org/10.1016/0304-3975(76)90059-1.
  16. Bas A. M. van Geffen, Bart M. P. Jansen, Arnoud A. W. M. de Kroon, and Rolf Morel. Lower Bounds for Dynamic Programming on Planar Graphs of Bounded Cutwidth. CoRR, abs/1806.10513, 2018. URL: http://arxiv.org/abs/1806.10513.
  17. Archontia C. Giannopoulou, Michal Pilipczuk, Jean-Florent Raymond, Dimitrios M. Thilikos, and Marcin Wrochna. Cutwidth: Obstructions and Algorithmic Aspects. In Proc. 11th IPEC, volume 63 of LIPIcs, pages 15:1-15:13, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.IPEC.2016.15.
  18. Russel Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1727.
  19. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which Problems Have Strongly Exponential Complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
  20. Bart M. P. Jansen and Jules J. H. M. Wulms. Lower Bounds for Protrusion Replacement by Counting Equivalence Classes. In Jiong Guo and Danny Hermelin, editors, Proc. 11th IPEC, volume 63 of LIPIcs, pages 17:1-17:12. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.IPEC.2016.17.
  21. Bart M.P. Jansen and Jesper Nederlof. Computing the Chromatic Number Using Graph Decompositions via Matrix Rank. In Proc. 26th ESA, 2018. In press. Google Scholar
  22. Ephraim Korach and Nir Solel. Tree-Width, Path-Width, and Cutwidth. Discrete Applied Mathematics, 43(1):97-101, 1993. URL: http://dx.doi.org/10.1016/0166-218X(93)90171-J.
  23. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known algorithms on graphs of bounded treewidth are probably optimal. In Proc. 22nd SODA, pages 777-789, 2011. URL: http://dx.doi.org/10.1137/1.9781611973082.61.
  24. Dániel Marx. The Square Root Phenomenon in Planar Graphs. In Michael R. Fellows, Xuehou Tan, and Binhai Zhu, editors, Proc. 3rd FAW-AAIM, volume 7924 of Lecture Notes in Computer Science, page 1. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-642-38756-2_1.
  25. Michal Pilipczuk. Problems Parameterized by Treewidth Tractable in Single Exponential Time: A Logical Approach. In Proc. 36th MFCS, pages 520-531, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22993-0_47.
  26. Dimitrios M. Thilikos, Maria J. Serna, and Hans L. Bodlaender. Cutwidth I: A linear time fixed parameter algorithm. J. Algorithms, 56(1):1-24, 2005. URL: http://dx.doi.org/10.1016/j.jalgor.2004.12.001.
  27. Dimitrios M. Thilikos, Maria J. Serna, and Hans L. Bodlaender. Cutwidth II: Algorithms for partial w-trees of bounded degree. J. Algorithms, 56(1):25-49, 2005. URL: http://dx.doi.org/10.1016/j.jalgor.2004.12.003.
  28. Johan M. M. van Rooij, Hans L. Bodlaender, and Peter Rossmanith. Dynamic Programming on Tree Decompositions Using Generalised Fast Subset Convolution. In Proc. 17th ESA, pages 566-577, 2009. URL: http://dx.doi.org/10.1007/978-3-642-04128-0_51.
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 About DROPS Imprint Privacy Contact