Reconfiguration of the Union of Arborescences

Authors Yusuke Kobayashi , Ryoga Mahara , Tamás Schwarcz



PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2023.48.pdf
  • Filesize: 0.75 MB
  • 14 pages

Document Identifiers

Author Details

Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Ryoga Mahara
  • Department of Mathematical Informatics, University of Tokyo, Japan
Tamás Schwarcz
  • MTA-ELTE Momentum Matroid Optimization Research Group, Department of Operations Research, ELTE Eötvös Loránd University, Budapest, Hungary

Acknowledgements

The authors thank members of the project "Fusion of Computer Science, Engineering and Mathematics Approaches for Expanding Combinatorial Reconfiguration" for discussion on this topic. The authors are grateful to András Frank for bringing the paper [Francisco Barahona and William R Pulleyblank, 1987] to their attention.

Cite As Get BibTex

Yusuke Kobayashi, Ryoga Mahara, and Tamás Schwarcz. Reconfiguration of the Union of Arborescences. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 48:1-48:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ISAAC.2023.48

Abstract

An arborescence in a digraph is an acyclic arc subset in which every vertex except a root has exactly one incoming arc. In this paper, we show the reconfigurability of the union of k arborescences for fixed k in the following sense: for any pair of arc subsets that can be partitioned into k arborescences, one can be transformed into the other by exchanging arcs one by one so that every intermediate arc subset can also be partitioned into k arborescences. This generalizes the result by Ito et al. (2023), who showed the case with k = 1. Since the union of k arborescences can be represented as a common matroid basis of two matroids, our result gives a new non-trivial example of matroid pairs for which two common bases are always reconfigurable to each other.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Combinatorial optimization
Keywords
  • Arborescence packing
  • common matroid basis
  • combinatorial reconfiguration

Metrics

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

References

  1. Francisco Barahona and William R Pulleyblank. Exact arborescences, matchings and cycles. Discrete Applied Mathematics, 16(2):91-99, 1987. URL: https://doi.org/10.1016/0166-218x(87)90067-9.
  2. Kristóf Bérczi and András Frank. Variations for Lovász’ submodular ideas. In Building Bridges, pages 137-164, 2010. URL: https://doi.org/10.1007/978-3-540-85221-6_4.
  3. Kristóf Bérczi and Tamás Schwarcz. Exchange distance of basis pairs in split matroids. arXiv preprint, 2022. URL: https://arxiv.org/abs/2203.01779.
  4. Joseph E. Bonin. Basis-exchange properties of sparse paving matroids. Advances in Applied Mathematics, 50(1):6-15, 2013. URL: https://doi.org/10.1016/j.aam.2011.05.006.
  5. Nicolas Bousquet, Takehiro Ito, Yusuke Kobayashi, Haruka Mizuta, Paul Ouvrard, Akira Suzuki, and Kunihiro Wasa. Reconfiguration of spanning trees with many or few leaves. In 28th Annual European Symposium on Algorithms (ESA 2020), pages 24:1-24:15, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.24.
  6. Nicolas Bousquet, Takehiro Ito, Yusuke Kobayashi, Haruka Mizuta, Paul Ouvrard, Akira Suzuki, and Kunihiro Wasa. Reconfiguration of spanning trees with degree constraint or diameter constraint. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022), pages 15:1-15:21, 2022. URL: https://doi.org/10.4230/LIPIcs.STACS.2022.15.
  7. Raul Cordovil and M. Leonor Moreira. Bases-cobases graphs and polytopes of matroids. Combinatorica, 13(2):157-165, 1993. URL: https://doi.org/10.1007/bf01303201.
  8. Olivier Durand de Gevigney, Viet-Hang Nguyen, and Zoltán Szigeti. Matroid-based packing of arborescences. SIAM Journal on Discrete Mathematics, 27(1):567-574, 2013. URL: https://doi.org/10.1137/120883761.
  9. Jack Edmonds. Edge-disjoint branchings. In Combinatorial algorithms, Courant Computer Science Symposium 9, pages 91-96. Algorithmics Press, New York, 1973. Google Scholar
  10. Martin Farber, Bruce Richter, and Herbert Shank. Edge-disjoint spanning trees: A connectedness theorem. Journal of Graph Theory, 9(3):319-324, 1985. URL: https://doi.org/10.1002/jgt.3190090303.
  11. Quentin Fortier, Csaba Király, Marion Léonard, Zoltán Szigeti, and Alexandre Talon. Old and new results on packing arborescences in directed hypergraphs. Discrete Applied Mathematics, 242:26-33, 2018. URL: https://doi.org/10.1016/j.dam.2017.11.004.
  12. András Frank. On disjoint trees and arborescences. In Algebraic Methods in Graph Theory, pages 159-169. North-Holland, Amsterdam, 1978. Google Scholar
  13. András Frank, Tamás Király, and Zoltán Király. On the orientation of graphs and hypergraphs. Discrete Applied Mathematics, 131(2):385-400, 2003. URL: https://doi.org/10.1016/S0166-218X(02)00462-6.
  14. Satoru Fujishige. A note on disjoint arborescences. Combinatorica, 30(2):247-252, 2010. URL: https://doi.org/10.1007/s00493-010-2518-y.
  15. Hui Gao and Daqing Yang. Packing of maximal independent mixed arborescences. Discrete Applied Mathematics, 289:313-319, 2021. URL: https://doi.org/10.1016/j.dam.2020.11.009.
  16. 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. Theoretical Computer Science, 412(12-14):1054-1065, 2011. URL: https://doi.org/10.1016/j.tcs.2010.12.005.
  17. Takehiro Ito, Yuni Iwamasa, Yasuaki Kobayashi, Yu Nakahata, Yota Otachi, and Kunihiro Wasa. Reconfiguring (non-spanning) arborescences. Theoretical Computer Science, 943:131-141, 2023. URL: https://doi.org/10.1016/j.tcs.2022.12.007.
  18. Takehiro Ito, Jun Kawahara, Yu Nakahata, Takehide Soh, Akira Suzuki, Junichi Teruyama, and Takahisa Toda. ZDD-based algorithmic framework for solving shortest reconfiguration problems. In Andre A. Cire, editor, Integration of Constraint Programming, Artificial Intelligence, and Operations Research, pages 167-183. Springer, Cham, Switzerland, 2023. URL: https://doi.org/10.1007/978-3-031-33271-5_12.
  19. Naoyuki Kamiyama, Naoki Katoh, and Atsushi Takizawa. Arc-disjoint in-trees in directed graphs. Combinatorica, 29(2):197-214, 2009. URL: https://doi.org/10.1007/s00493-009-2428-z.
  20. Csaba Király. On maximal independent arborescence packing. SIAM Journal on Discrete Mathematics, 30(4):2107-2114, 2016. URL: https://doi.org/10.1137/130938396.
  21. Michał Lasoń and Mateusz Michałek. On the toric ideal of a matroid. Advances in Mathematics, 259:1-12, 2014. URL: https://doi.org/10.1016/j.aim.2014.03.004.
  22. László Lovász. On two minimax theorems in graph. Journal of Combinatorial Theory, Series B, 21(2):96-103, 1976. URL: https://doi.org/10.1016/0095-8956(76)90049-6.
  23. Tatsuya Matsuoka and Shin-ichi Tanigawa. On reachability mixed arborescence packing. Discrete Optimization, 32:1-10, 2019. URL: https://doi.org/10.1016/j.disopt.2018.10.002.
  24. Naomi Nishimura. Introduction to reconfiguration. Algorithms, 11(4):52, 2018. URL: https://doi.org/10.3390/a11040052.
  25. James Oxley. Matroid Theory, volume 21 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, second edition, 2011. Google Scholar
  26. Alexander Schrijver. Combinatorial Optimization: Polyhedra and Efficiency, volume 24 of Algorithms and Combinatorics. Springer-Verlag, Berlin, 2003. Google Scholar
  27. Jan van den Heuvel. 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, Cambridge, UK, 2013. URL: https://doi.org/10.1017/CBO9781139506748.005.
  28. Neil L. White. A unique exchange property for bases. Linear Algebra and its Applications, 31:81-91, 1980. URL: https://doi.org/10.1016/0024-3795(80)90209-8.
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