Token Sliding on Split Graphs

Authors Rémy Belmonte , Eun Jung Kim, Michael Lampis , Valia Mitsou, Yota Otachi , Florian Sikora



PDF
Thumbnail PDF

File

LIPIcs.STACS.2019.13.pdf
  • Filesize: 0.49 MB
  • 17 pages

Document Identifiers

Author Details

Rémy Belmonte
  • University of Electro-Communications, Chofu, Tokyo, 182-8585, Japan
Eun Jung Kim
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France
Michael Lampis
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France
Valia Mitsou
  • Université Paris-Diderot, IRIF, CNRS, 75205, Paris, France
Yota Otachi
  • Kumamoto University, Kumamoto, 860-8555, Japan
Florian Sikora
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France

Cite AsGet BibTex

Rémy Belmonte, Eun Jung Kim, Michael Lampis, Valia Mitsou, Yota Otachi, and Florian Sikora. Token Sliding on Split Graphs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.STACS.2019.13

Abstract

We consider the complexity of the Independent Set Reconfiguration problem under the Token Sliding rule. In this problem we are given two independent sets of a graph and are asked if we can transform one to the other by repeatedly exchanging a vertex that is currently in the set with one of its neighbors, while maintaining the set independent. Our main result is to show that this problem is PSPACE-complete on split graphs (and hence also on chordal graphs), thus resolving an open problem in this area. We then go on to consider the c-Colorable Reconfiguration problem under the same rule, where the constraint is now to maintain the set c-colorable at all times. As one may expect, a simple modification of our reduction shows that this more general problem is PSPACE-complete for all fixed c >= 1 on chordal graphs. Somewhat surprisingly, we show that the same cannot be said for split graphs: we give a polynomial time (n^{O(c)}) algorithm for all fixed values of c, except c=1, for which the problem is PSPACE-complete. We complement our algorithm with a lower bound showing that c-Colorable Reconfiguration is W[2]-hard on split graphs parameterized by c and the length of the solution, as well as a tight ETH-based lower bound for both parameters.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • reconfiguration
  • independent set
  • split graph

Metrics

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

References

  1. Marthe Bonamy and Nicolas Bousquet. Token Sliding on Chordal Graphs. In WG 2017, volume 10520 of Lecture Notes in Computer Science, pages 127-139, 2017. URL: http://dx.doi.org/10.1007/978-3-319-68705-6_10.
  2. Paul S. Bonsma, Marcin Kaminski, and Marcin Wrochna. Reconfiguring Independent Sets in Claw-Free Graphs. In SWAT, volume 8503 of Lecture Notes in Computer Science, pages 86-97. Springer, 2014. Google Scholar
  3. Nicolas Bousquet, Arnaud Mary, and Aline Parreau. Token Jumping in Minor-Closed Classes. In FCT, volume 10472 of Lecture Notes in Computer Science, pages 136-149. Springer, 2017. Google Scholar
  4. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  5. Erik D. Demaine, Martin L. Demaine, Eli Fox-Epstein, Duc A. Hoang, Takehiro Ito, Hirotaka Ono, Yota Otachi, Ryuhei Uehara, and Takeshi Yamada. Linear-time algorithm for sliding tokens on trees. Theor. Comput. Sci., 600:132-142, 2015. URL: http://dx.doi.org/10.1016/j.tcs.2015.07.037.
  6. Eli Fox-Epstein, Duc A. Hoang, Yota Otachi, and Ryuhei Uehara. Sliding Token on Bipartite Permutation Graphs. In ISAAC, volume 9472 of Lecture Notes in Computer Science, pages 237-247. Springer, 2015. Google Scholar
  7. Robert A. Hearn. Games, puzzles, and computation. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 2006. URL: http://hdl.handle.net/1721.1/37913.
  8. Robert A. Hearn and Erik D. Demaine. PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theor. Comput. Sci., 343(1-2):72-96, 2005. URL: http://dx.doi.org/10.1016/j.tcs.2005.05.008.
  9. Robert A. Hearn and Erik D. Demaine. Games, puzzles and computation. A K Peters, 2009. Google Scholar
  10. Jan . The complexity of change. In Simon R. Blackburn, Stefanie Gerke, and Mark Wildon, editors, Surveys in Combinatorics 2013, volume 409 of London Mathematical Society Lecture Note Series, pages 127-160. Cambridge University Press, 2013. URL: http://dx.doi.org/10.1017/CBO9781139506748.005.
  11. Duc A. Hoang and Ryuhei Uehara. Sliding Tokens on a Cactus. In ISAAC, volume 64 of LIPIcs, pages 37:1-37:26. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  12. Takehiro Ito, Erik D. Demaine, Nicholas J. A. Harvey, Christos H. Papadimitriou, Martha Sideri, Ryuhei Uehara, and Yushi Uno. On the complexity of reconfiguration problems. Theor. Comput. Sci., 412(12-14):1054-1065, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2010.12.005.
  13. Takehiro Ito, Marcin Kaminski, Hirotaka Ono, Akira Suzuki, Ryuhei Uehara, and Katsuhisa Yamanaka. On the Parameterized Complexity for Token Jumping on Graphs. In TAMC, volume 8402 of Lecture Notes in Computer Science, pages 341-351. Springer, 2014. Google Scholar
  14. Takehiro Ito, Marcin Jakub Kaminski, and Hirotaka Ono. Fixed-Parameter Tractability of Token Jumping on Planar Graphs. In ISAAC, volume 8889 of Lecture Notes in Computer Science, pages 208-219. Springer, 2014. Google Scholar
  15. Takehiro Ito and Yota Otachi. Reconfiguration of Colorable Sets in Classes of Perfect Graphs. In SWAT, volume 101 of LIPIcs, pages 27:1-27:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  16. Marcin Kamiński, Paul Medvedev, and Martin Milanič. Complexity of independent set reconfigurability problems. Theor. Comput. Sci., 439:9-15, 2012. URL: http://dx.doi.org/10.1016/j.tcs.2012.03.004.
  17. Daniel Lokshtanov and Amer E. Mouawad. The complexity of independent set reconfiguration on bipartite graphs. In SODA 2018, pages 185-195, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.13.
  18. Amer E. Mouawad, Naomi Nishimura, Venkatesh Raman, Narges Simjour, and Akira Suzuki. On the Parameterized Complexity of Reconfiguration Problems. Algorithmica, 78(1):274-297, 2017. Google Scholar
  19. Naomi Nishimura. Introduction to Reconfiguration. Algorithms, 11(4):52, 2018. URL: http://dx.doi.org/10.3390/a11040052.
  20. Christos H. Papadimitriou. Computational complexity. Addison-Wesley, 1994. Google Scholar
  21. Douglas B. West. Introduction to graph theory. Prentice Hall, Upper Saddle River, 2nd edition, 2001. Google Scholar
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