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Token Sliding Reconfiguration on DAGs

Authors Jona Dirks , Alexandre Vigny

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LIPIcs.MFCS.2025.41.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Fixed parameter tractability
Keywords
  • Graph theory
  • FPT algorithms
  • Reconfiguration
  • Independent Sets

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Abstract

Given a graph G and two independent sets of same size, the Independent Set Reconfiguration Problem under token sliding asks whether one can, in a step by step manner, transform the first independent set into the second one. In each step we must preserve the condition of independence. Further, referring to solution vertices as tokens, we are only permitted to slide a token along an edge. Until the recent work of Ito et al. [Ito et al. MFCS 2022] this problem was only considered on undirected graphs. In this work, we study reconfiguration under token sliding focusing on DAGs. 
We present a complete dichotomy of intractability in regard to the depth of the DAG, by proving that this problem is NP-complete for DAGs of depth 3 and W[1]-hard for depth 4 when parameterized by the number of tokens k, and that these bounds are tight. Further, we prove that it is fixed parameter tractable on DAGs parameterized by the combination of treewidth and k. We show that this result applies to undirected graphs, when the number of times a token can visit a vertex is restricted.

Cite As Get BibTex

Jona Dirks and Alexandre Vigny. Token Sliding Reconfiguration on DAGs. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 41:1-41:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.41

Author Details

Jona Dirks
  • Université Clermont Auvergne, Clermont Auvergne INP, LIMOS, Clermont-Ferrand, France
Alexandre Vigny
  • Université Clermont Auvergne, Clermont Auvergne INP, LIMOS, Clermont-Ferrand, France

Funding

This work benefited from state aid managed by the ANR under France 2030 referenced ANR-23-IACL-0006.

References

  1. Niranka Banerjee, Christian Engels, and Duc A. Hoang. Directed token sliding, 2024. URL: https://doi.org/10.48550/arXiv.2411.16149.
  2. Valentin Bartier, Nicolas Bousquet, Jihad Hanna, Amer E. Mouawad, and Sebastian Siebertz. Token sliding on graphs of girth five. Algorithmica, 86(2):638-655, 2024. URL: https://doi.org/10.1007/S00453-023-01181-5.
  3. Valentin Bartier, Nicolas Bousquet, and Amer E. Mouawad. Galactic token sliding. J. Comput. Syst. Sci., 136:220-248, 2023. URL: https://doi.org/10.1016/J.JCSS.2023.03.008.
  4. Hans L. Bodlaender, Carla Groenland, and Céline M. F. Swennenhuis. Parameterized complexities of dominating and independent set reconfiguration. In Intl. Symp. on Parameterized and Exact Computation (IPEC), volume 214 of LIPIcs, pages 9:1-9:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.IPEC.2021.9.
  5. Marthe Bonamy and Nicolas Bousquet. Token sliding on chordal graphs. In Workshop on Graph-Theoretic Concepts in Computer Science (WG), volume 10520 of Lecture Notes in Computer Science, pages 127-139. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-68705-6_10.
  6. Nicolas Bousquet, Amer E. Mouawad, Naomi Nishimura, and Sebastian Siebertz. A survey on the parameterized complexity of reconfiguration problems. Comput. Sci. Rev., 53:100663, 2024. URL: https://doi.org/10.1016/J.COSREV.2024.100663.
  7. 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.
  8. Erik D. Demaine, Martin L. Demaine, Eli Fox-Epstein, Duc A. Hoang, Takehiro Ito, Hirotaka Ono, Yota Otachi, Ryuhei Uehara, and Takeshi Yamada. Polynomial-time algorithm for sliding tokens on trees. In Intl. Symp. on Algorithms and Computation (ISAAC), volume 8889 of Lecture Notes in Computer Science, pages 389-400. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-13075-0_31.
  9. Tatsuya Gima, Takehiro Ito, Yasuaki Kobayashi, and Yota Otachi. Algorithmic meta-theorems for combinatorial reconfiguration revisited. Algorithmica, 86(11):3395-3424, 2024. URL: https://doi.org/10.1007/S00453-024-01261-0.
  10. Arash Haddadan, Takehiro Ito, Amer E. Mouawad, Naomi Nishimura, Hirotaka Ono, Akira Suzuki, and Youcef Tebbal. The complexity of dominating set reconfiguration. Theor. Comput. Sci., 651:37-49, 2016. URL: https://doi.org/10.1016/J.TCS.2016.08.016.
  11. Takehiro Ito, Yuni Iwamasa, Yasuaki Kobayashi, Yu Nakahata, Yota Otachi, Masahiro Takahashi, and Kunihiro Wasa. Independent set reconfiguration on directed graphs. In Intl. Symp. on Mathematical Foundations of Computer Science (MFCS), volume 241 of LIPIcs, pages 58:1-58:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.MFCS.2022.58.
  12. Wm. Woolsey Johnson and William E. Story. Notes on the "15" puzzle. American Journal of Mathematics, 2(4):397-404, 1879. URL: http://www.jstor.org/stable/2369492.
  13. Daniel Lokshtanov and Amer E. Mouawad. The complexity of independent set reconfiguration on bipartite graphs. ACM Trans. Algorithms, 15(1):7:1-7:19, 2019. URL: https://doi.org/10.1145/3280825.
  14. 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. URL: https://doi.org/10.1007/S00453-016-0159-2.
  15. Amer E. Mouawad, Naomi Nishimura, Venkatesh Raman, and Marcin Wrochna. Reconfiguration over tree decompositions. In Intl. Symp. on Parameterized and Exact Computation (IPEC), volume 8894 of Lecture Notes in Computer Science, pages 246-257. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-13524-3_21.
  16. Naomi Nishimura. Introduction to reconfiguration. Algorithms, 11(4):52, 2018. URL: https://doi.org/10.3390/A11040052.
  17. Sebastian Siebertz. Reconfiguration on nowhere dense graph classes. Electron. J. Comb., 25(3):3, 2018. URL: https://doi.org/10.37236/7458.
  18. Marcin Wrochna. Reconfiguration in bounded bandwidth and tree-depth, 2018. URL: https://doi.org/10.1016/J.JCSS.2017.11.003.
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