Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space

Authors Falko Hegerfeld , Stefan Kratsch



PDF
Thumbnail PDF

File

LIPIcs.STACS.2020.29.pdf
  • Filesize: 0.64 MB
  • 16 pages

Document Identifiers

Author Details

Falko Hegerfeld
  • Humboldt-Universität zu Berlin, Germany
Stefan Kratsch
  • Humboldt-Universität zu Berlin, Germany

Cite AsGet BibTex

Falko Hegerfeld and Stefan Kratsch. Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.STACS.2020.29

Abstract

A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time ?^*(2^{?(twlog tw)}). Using their inspired Cut&Count technique, they obtained ?^*(α^tw) time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely believed to be unavoidable. In contrast, for the slightly larger parameter called treedepth, there are already several examples of matching the time bounds obtained for treewidth, but using only polynomial space. Nevertheless, this has remained open for connectivity problems. In the present work, we close this knowledge gap by applying the Cut&Count technique to graphs of small treedepth. While the general idea is unchanged, we have to design novel procedures for counting consistently cut solution candidates using only polynomial space. Concretely, we obtain time ?^*(3^d) and polynomial space for Connected Vertex Cover, Feedback Vertex Set, and Steiner Tree on graphs of treedepth d. Similarly, we obtain time ?^*(4^d) and polynomial space for Connected Dominating Set and Connected Odd Cycle Transversal.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Paths and connectivity problems
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Parameterized Complexity
  • Connectivity
  • Treedepth
  • Cut&Count
  • Polynomial Space

Metrics

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

References

  1. Jochen Alber and Rolf Niedermeier. Improved tree decomposition based algorithms for domination-like problems. In Sergio Rajsbaum, editor, LATIN 2002: Theoretical Informatics, 5th Latin American Symposium, Cancun, Mexico, April 3-6, 2002, Proceedings, volume 2286 of Lecture Notes in Computer Science, pages 613-628. Springer, 2002. URL: https://doi.org/10.1007/3-540-45995-2_52.
  2. Mahdi Belbasi and Martin Fürer. A space-efficient parameterized algorithm for the Hamiltonian cycle problem by dynamic algebraization. In René van Bevern and Gregory Kucherov, editors, Computer Science - Theory and Applications - 14th International Computer Science Symposium in Russia, CSR 2019, Novosibirsk, Russia, July 1-5, 2019, Proceedings, volume 11532 of Lecture Notes in Computer Science, pages 38-49. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-19955-5_4.
  3. Benjamin Bergougnoux and Mamadou Moustapha Kanté. Fast exact algorithms for some connectivity problems parameterized by clique-width. Theor. Comput. Sci., 782:30-53, 2019. URL: https://doi.org/10.1016/j.tcs.2019.02.030.
  4. Benjamin Bergougnoux and Mamadou Moustapha Kanté. More applications of the d-neighbor equivalence: Connectivity and acyclicity constraints. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany., volume 144 of LIPIcs, pages 17:1-17:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.17.
  5. 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: https://doi.org/10.1016/j.ic.2014.12.008.
  6. Hans L. Bodlaender, Jitender S. Deogun, Klaus Jansen, Ton Kloks, Dieter Kratsch, Haiko Müller, and Zsolt Tuza. Rankings of graphs. SIAM J. Discrete Math., 11(1):168-181, 1998. URL: https://doi.org/10.1137/S0895480195282550.
  7. Hans L. Bodlaender, John R. Gilbert, Hjálmtyr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms, 18(2):238-255, 1995. URL: https://doi.org/10.1006/jagm.1995.1009.
  8. Li-Hsuan Chen, Felix Reidl, Peter Rossmanith, and Fernando Sánchez Villaamil. Width, depth, and space: Tradeoffs between branching and dynamic programming. Algorithms, 11(7):98, 2018. URL: https://doi.org/10.3390/a11070098.
  9. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and computation, 85(1):12-75, 1990. Google Scholar
  10. Marek Cygan, Fedor V. Fomin, Lukasz 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.
  11. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michał Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 150-159. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.23.
  12. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: https://doi.org/10.1007/978-1-4471-5559-1.
  13. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. URL: https://doi.org/10.1007/3-540-29953-X.
  14. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Representative sets of product families. 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 443-454. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-44777-2_37.
  15. Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Efficient computation of representative sets with applications in parameterized and exact algorithms. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 142-151. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.10.
  16. Markus Frick and Martin Grohe. The complexity of first-order and monadic second-order logic revisited. Ann. Pure Appl. Logic, 130(1-3):3-31, 2004. URL: https://doi.org/10.1016/j.apal.2004.01.007.
  17. Martin Fürer and Huiwen Yu. Space saving by dynamic algebraization based on tree-depth. Theory Comput. Syst., 61(2):283-304, 2017. URL: https://doi.org/10.1007/s00224-017-9751-3.
  18. Falko Hegerfeld and Stefan Kratsch. Solving connectivity problems parameterized by treedepth in single-exponential time and polynomial space. arXiv e-prints, page arXiv:2001.05364, January 2020. URL: http://arxiv.org/abs/2001.05364.
  19. Meir Katchalski, William McCuaig, and Suzanne M. Seager. Ordered colourings. Discrete Mathematics, 142(1-3):141-154, 1995. URL: https://doi.org/10.1016/0012-365X(93)E0216-Q.
  20. Ton Kloks. Treewidth, Computations and Approximations, volume 842 of Lecture Notes in Computer Science. Springer, 1994. URL: https://doi.org/10.1007/BFb0045375.
  21. Jason Li and Jesper Nederlof. Detecting feedback vertex sets of size k in ?^*(2.7^k) time. CoRR, abs/1906.12298, 2019. URL: http://arxiv.org/abs/1906.12298.
  22. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known algorithms on graphs of bounded treewidth are probably optimal. ACM Trans. Algorithms, 14(2):13:1-13:30, 2018. URL: https://doi.org/10.1145/3170442.
  23. Daniel Lokshtanov and Jesper Nederlof. Saving space by algebraization. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 321-330. ACM, 2010. URL: https://doi.org/10.1145/1806689.1806735.
  24. Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):105-113, 1987. URL: https://doi.org/10.1007/BF02579206.
  25. Jaroslav Nešetřil and Patrice Ossona de Mendez. Tree-depth, subgraph coloring and homomorphism bounds. Eur. J. Comb., 27(6):1022-1041, 2006. URL: https://doi.org/10.1016/j.ejc.2005.01.010.
  26. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-27875-4.
  27. Jaroslav Nešetřil and Patrice Ossona de Mendez. On low tree-depth decompositions. Graphs and Combinatorics, 31(6):1941-1963, 2015. URL: https://doi.org/10.1007/s00373-015-1569-7.
  28. Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford University Press, 2006. URL: https://doi.org/10.1093/ACPROF:OSO/9780198566076.001.0001.
  29. Michał Pilipczuk and Marcin Wrochna. On space efficiency of algorithms working on structural decompositions of graphs. TOCT, 9(4):18:1-18:36, 2018. URL: https://doi.org/10.1145/3154856.
  30. Willem J. A. Pino, Hans L. Bodlaender, and Johan M. M. van Rooij. Cut and count and representative sets on branch decompositions. In Jiong Guo and Danny Hermelin, editors, 11th International Symposium on Parameterized and Exact Computation, IPEC 2016, August 24-26, 2016, Aarhus, Denmark, volume 63 of LIPIcs, pages 27:1-27:12. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.IPEC.2016.27.
  31. Jan Arne Telle and Andrzej Proskurowski. Practical algorithms on partial k-trees with an application to domination-like problems. In Frank K. H. A. Dehne, Jörg-Rüdiger Sack, Nicola Santoro, and Sue Whitesides, editors, Algorithms and Data Structures, Third Workshop, WADS '93, Montréal, Canada, August 11-13, 1993, Proceedings, volume 709 of Lecture Notes in Computer Science, pages 610-621. Springer, 1993. URL: https://doi.org/10.1007/3-540-57155-8_284.
  32. Johan M. M. van Rooij, Hans L. Bodlaender, and Peter Rossmanith. Dynamic programming on tree decompositions using generalised fast subset convolution. In Amos Fiat and Peter Sanders, editors, Algorithms - ESA 2009, 17th Annual European Symposium, Copenhagen, Denmark, September 7-9, 2009. Proceedings, volume 5757 of Lecture Notes in Computer Science, pages 566-577. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-04128-0_51.
  33. Johan M. M. van Rooij, Hans L. Bodlaender, Erik Jan van Leeuwen, Peter Rossmanith, and Martin Vatshelle. Fast dynamic programming on graph decompositions. CoRR, abs/1806.01667, 2018. URL: http://arxiv.org/abs/1806.01667.
  34. Gerhard J. Woeginger. Space and time complexity of exact algorithms: Some open problems (invited talk). In Rodney G. Downey, Michael R. Fellows, and Frank K. H. A. Dehne, editors, Parameterized and Exact Computation, First International Workshop, IWPEC 2004, Bergen, Norway, September 14-17, 2004, Proceedings, volume 3162 of Lecture Notes in Computer Science, pages 281-290. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-28639-4_25.
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