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Rerouting Planar Curves and Disjoint Paths

Authors Takehiro Ito , Yuni Iwamasa , Naonori Kakimura , Yusuke Kobayashi , Shun-ichi Maezawa , Yuta Nozaki , Yoshio Okamoto , Kenta Ozeki

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  • Theory of computation → Design and analysis of algorithms
Keywords
  • Disjoint paths
  • combinatorial reconfiguration
  • planar graphs

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Abstract

In this paper, we consider a transformation of k disjoint paths in a graph. For a graph and a pair of k disjoint paths 𝒫 and 𝒬 connecting the same set of terminal pairs, we aim to determine whether 𝒫 can be transformed to 𝒬 by repeatedly replacing one path with another path so that the intermediates are also k disjoint paths. The problem is called Disjoint Paths Reconfiguration. We first show that Disjoint Paths Reconfiguration is PSPACE-complete even when k = 2. On the other hand, we prove that, when the graph is embedded on a plane and all paths in 𝒫 and 𝒬 connect the boundaries of two faces, Disjoint Paths Reconfiguration can be solved in polynomial time. The algorithm is based on a topological characterization for rerouting curves on a plane using the algebraic intersection number. We also consider a transformation of disjoint s-t paths as a variant. We show that the disjoint s-t paths reconfiguration problem in planar graphs can be determined in polynomial time, while the problem is PSPACE-complete in general.

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Takehiro Ito, Yuni Iwamasa, Naonori Kakimura, Yusuke Kobayashi, Shun-ichi Maezawa, Yuta Nozaki, Yoshio Okamoto, and Kenta Ozeki. Rerouting Planar Curves and Disjoint Paths. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 81:1-81:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.81

Author Details

Takehiro Ito
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Yuni Iwamasa
  • Graduate School of Informatics, Kyoto University, Japan
Naonori Kakimura
  • Faculty of Science and Technology, Keio University, Yokohama, Japan
Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Shun-ichi Maezawa
  • Department of Mathematics, Tokyo University of Science, Japan
Yuta Nozaki
  • Faculty of Environment and Information Sciences, Yokohama National University, Japan
  • SKCM, Hiroshima University, Japan
Yoshio Okamoto
  • Graduate School of Informatics and Engineering, The University of Electro-Communications, Tokyo, Japan
Kenta Ozeki
  • Faculty of Environment and Information Sciences, Yokohama National University, Japan

Funding

  • Ito, Takehiro: JSPS KAKENHI Grant Numbers JP18H04091, JP19K11814, JP20H05793.
  • Iwamasa, Yuni: JSPS KAKENHI Grant Numbers JP20K23323, JP20H05795, JP22K17854.
  • Kakimura, Naonori: JSPS KAKENHI Grant Numbers JP20H05795, JP21H03397, JP22H05001.
  • Kobayashi, Yusuke: JSPS KAKENHI Grant Numbers JP20K11692, JP20H05795, JP22H05001.
  • Maezawa, Shun-ichi: JSPS KAKENHI Grant Numbers JP20H05795, JP22K13956.
  • Nozaki, Yuta: JSPS KAKENHI Grant Numbers JP20H05795, JP20K14317, JP23K12974.
  • Okamoto, Yoshio: JSPS KAKENHI Grant Numbers JP20H05795, JP20K11670, JP23K10982.
  • Ozeki, Kenta: JSPS KAKENHI Grant Numbers JP18K03391, JP19H01803, JP20H05795, 23K03195.

Acknowledgements

We thank Naoyuki Kamiyama for the discussion and the anonymous referees for their helpful comments.

References

  1. Isolde Adler, Stavros G Kolliopoulos, Philipp Klaus Krause, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M Thilikos. Irrelevant vertices for the planar disjoint paths problem. Journal of Combinatorial Theory, Series B, 122:815-843, 2017. URL: https://doi.org/10.1016/j.jctb.2016.10.001.
  2. Saeed Akhoondian Amiri, Szymon Dudycz, Mahmoud Parham, Stefan Schmid, and Sebastian Wiederrecht. On polynomial-time congestion-free software-defined network updates. In 2019 IFIP Networking Conference, Networking 2019, Warsaw, Poland, May 20-22, 2019, pages 1-9. IEEE, 2019. URL: https://doi.org/10.23919/IFIPNetworking.2019.8816833.
  3. Saeed Akhoondian Amiri, Szymon Dudycz, Stefan Schmid, and Sebastian Wiederrecht. Congestion-free rerouting of flows on DAGs. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 143:1-143:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.143.
  4. John Asplund, Kossi D. Edoh, Ruth Haas, Yulia Hristova, Beth Novick, and Brett Werner. Reconfiguration graphs of shortest paths. Discret. Math., 341(10):2938-2948, 2018. URL: https://doi.org/10.1016/j.disc.2018.07.007.
  5. John Asplund and Brett Werner. Classification of reconfiguration graphs of shortest path graphs with no induced 4-cycles. Discret. Math., 343(1):111640, 2020. URL: https://doi.org/10.1016/j.disc.2019.111640.
  6. Paul S. Bonsma. The complexity of rerouting shortest paths. Theor. Comput. Sci., 510:1-12, 2013. URL: https://doi.org/10.1016/j.tcs.2013.09.012.
  7. Paul S. Bonsma. Rerouting shortest paths in planar graphs. Discret. Appl. Math., 231:95-112, 2017. URL: https://doi.org/10.1016/j.dam.2016.05.024.
  8. Glencora Borradaile, Amir Nayyeri, and Farzad Zafarani. Towards single face shortest vertex-disjoint paths in undirected planar graphs. In Proceedings of 23rd Annual European Symposium on Algorithms (ESA), volume 9294 of Lecture Notes in Computer Science, pages 227-238, 2015. URL: https://doi.org/10.1007/978-3-662-48350-3_20.
  9. Éric Colin de Verdière and Alexander Schrijver. Shortest vertex-disjoint two-face paths in planar graphs. ACM Transactions on Algorithms, 7(2):1-12, 2011. URL: https://doi.org/10.1145/1921659.1921665.
  10. Marek Cygan, Dániel Marx, Marcin Pilipczuk, and Michal Pilipczuk. The planar directed k-vertex-disjoint paths problem is fixed-parameter tractable. In 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2013), pages 197-206, 2013. URL: https://doi.org/10.1109/FOCS.2013.29.
  11. Samir Datta, Siddharth Iyer, Raghav Kulkarni, and Anish Mukherjee. Shortest k-disjoint paths via determinants. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018), volume 122, pages 19:1-19:21, 2018. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2018.19.
  12. Julian Enoch, Kyle Fox, Dor Mesica, and Shay Mozes. A faster algorithm for maximum flow in directed planar graphs with vertex capacities. In Hee-Kap Ahn and Kunihiko Sadakane, editors, 32nd International Symposium on Algorithms and Computation, ISAAC 2021, December 6-8, 2021, Fukuoka, Japan, volume 212 of LIPIcs, pages 72:1-72:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.72.
  13. Benson Farb and Dan Margalit. A primer on mapping class groups, volume 49 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 2012. Google Scholar
  14. Lester R. Ford and Delbert R. Fulkerson. Maximal flow through a network. Canadian Journal of Mathematics, 8:399-404, 1956. URL: https://doi.org/10.4153/CJM-1956-045-5.
  15. Steven Fortune, John Hopcroft, and James Wyllie. The directed subgraph homeomorphism problem. Theoretical Computer Science, 10(2):111-121, 1980. URL: https://doi.org/10.1016/0304-3975(80)90009-2.
  16. András Frank. Packing paths, circuits, and cuts - A survey. In Paths, Flows, and VLSI-Layout, pages 47-100. Springer, 1990. Google Scholar
  17. Kshitij Gajjar, Agastya Vibhuti Jha, Manish Kumar, and Abhiruk Lahiri. Reconfiguring shortest paths in graphs. In Thirty-Sixth AAAI Conference on Artificial Intelligence, AAAI 2022, pages 9758-9766. AAAI Press, 2022. Google Scholar
  18. Refael Hassin. On multicommodity flows in planar graphs. Networks, 14(2):225-235, 1984. URL: https://doi.org/10.1002/net.3230140204.
  19. 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: https://doi.org/10.1016/j.tcs.2010.12.005.
  20. Takehiro Ito, Yuni Iwamasa, Naonori Kakimura, Yusuke Kobayashi, Shun ichi Maezawa, Yuta Nozaki, Yoshio Okamoto, and Kenta Ozeki. Rerouting planar curves and disjoint paths. arXiv:2210.11778, 2022. URL: https://doi.org/10.48550/arXiv.2210.11778.
  21. Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, and Yoshio Okamoto. Reconfiguration of maximum-weight b-matchings in a graph. J. Comb. Optim., 37(2):454-464, 2019. URL: https://doi.org/10.1007/s10878-018-0289-3.
  22. Takehiro Ito, Jun Kawahara, Yu Nakahata, Takehide Soh, Akira Suzuki, Junichi Teruyama, and Takahisa Toda. ZDD-based algorithmic framework for solving shortest reconfiguration problems. arXiv:2207.13959, 2022. URL: https://doi.org/10.48550/arXiv.2207.13959.
  23. Marcin Kaminski, Paul Medvedev, and Martin Milanic. Shortest paths between shortest paths. Theor. Comput. Sci., 412(39):5205-5210, 2011. URL: https://doi.org/10.1016/j.tcs.2011.05.021.
  24. Haim Kaplan and Yahav Nussbaum. Maximum flow in directed planar graphs with vertex capacities. Algorithmica, 61(1):174-189, 2011. URL: https://doi.org/10.1007/s00453-010-9436-7.
  25. Richard M Karp. On the computational complexity of combinatorial problems. Networks, 5(1):45-68, 1975. URL: https://doi.org/10.1002/net.1975.5.1.45.
  26. Samir Khuller and Joseph Naor. Flow in planar graphs with vertex capacities. Algorithmica, 11(3):200-225, 1994. URL: https://doi.org/10.1007/BF01240733.
  27. Yusuke Kobayashi and Kensuke Otsuki. Max-flow min-cut theorem and faster algorithms in a circular disk failure model. In 2014 IEEE Conference on Computer Communications (INFOCOM), pages 1635-1643, 2014. URL: https://doi.org/10.1109/INFOCOM.2014.6848100.
  28. Yusuke Kobayashi and Christian Sommer. On shortest disjoint paths in planar graphs. Discrete Optimization, 7(4):234-245, 2010. URL: https://doi.org/10.1016/j.disopt.2010.05.002.
  29. Mark R Kramer. The complexity of wire-routing and finding minimum area layouts for arbitrary VLSI circuits. Advances in Computing Research, 2:129-146, 1984. Google Scholar
  30. James F Lynch. The equivalence of theorem proving and the interconnection problem. ACM SIGDA Newsletter, 5(3):31-36, 1975. URL: https://doi.org/10.1145/1061425.1061430.
  31. Colin J. H. McDiarmid, Bruce A. Reed, Alexander Schrijver, and F. Bruce Shepherd. Non-interfering network flows. In Proceedings of the Third Scandinavian Workshop on Algorithm Theory (SWAT), pages 245-257, 1992. URL: https://doi.org/10.1007/3-540-55706-7_21.
  32. Colin J. H. McDiarmid, Bruce A. Reed, Alexander Schrijver, and F. Bruce Shepherd. Induced circuits in planar graphs. Journal of Combinatorial Theory, Series B, 60(2):169-176, 1994. URL: https://doi.org/10.1006/jctb.1994.1011.
  33. Karl Menger. Zur allgemeinen Kurventheorie. Fundamenta Mathematicae, 10:96-115, 1927. Google Scholar
  34. Naomi Nishimura. Introduction to reconfiguration. Algorithms, 11(4):52, 2018. URL: https://doi.org/10.3390/a11040052.
  35. Haruko Okamura. Multicommodity flows in graphs. Discrete Applied Mathematics, 6(1):55-62, 1983. URL: https://doi.org/10.1016/0166-218X(83)90100-2.
  36. Haruko Okamura and Paul D. Seymour. Multicommodity flows in planar graphs. Journal of Combinatorial Theory, Series B, 31(1):75-81, 1981. URL: https://doi.org/10.1016/S0095-8956(81)80012-3.
  37. Kensuke Otsuki, Yusuke Kobayashi, and Kazuo Murota. Improved max-flow min-cut algorithms in a circular disk failure model with application to a road network. Eur. J. Oper. Res., 248(2):396-403, 2016. URL: https://doi.org/10.1016/j.ejor.2015.07.035.
  38. Bruce Reed. Rooted routing in the plane. Discrete Applied Mathematics, 57:213-227, 1995. URL: https://doi.org/10.1016/0166-218X(94)00104-L.
  39. Bruce Reed, Neil Robertson, Alexander Schrijver, and Paul D. Seymour. Finding disjoint trees in planar graphs in linear time. In Contemporary Mathematics 147, pages 295-301. American Mathematical Society, 1993. URL: https://doi.org/10.1090/conm/147/01180.
  40. Neil Robertson and Paul D. Seymour. Graph minors. VI. Disjoint paths across a disc. Journal of Combinatorial Theory, Series B, 41(1):115-138, 1986. URL: https://doi.org/10.1016/0095-8956(86)90031-6.
  41. Neil Robertson and Paul D. Seymour. Graph minors. VII. Disjoint paths on a surface. Journal of Combinatorial Theory, Series B, 45(2):212-254, 1988. URL: https://doi.org/10.1016/0095-8956(88)90070-6.
  42. Neil Robertson and Paul D. Seymour. Graph minors. XIII. The disjoint paths problem. Journal of combinatorial theory, Series B, 63(1):65-110, 1995. URL: https://doi.org/10.1006/jctb.1995.1006.
  43. Alexander Schrijver. Finding k disjoint paths in a directed planar graph. SIAM Journal on Computing, 23(4):780-788, 1994. URL: https://doi.org/10.1137/S0097539792224061.
  44. Paul D. Seymour. Disjoint paths in graphs. Discrete Mathematics, 29:293-309, 1980. URL: https://doi.org/10.1016/0012-365X(80)90158-2.
  45. Yossi Shiloach. A polynomial solution to the undirected two paths problem. Journal of the ACM, 27(3):445-456, 1980. URL: https://doi.org/10.1145/322203.322207.
  46. Anand Srinivas and Eytan Modiano. Finding minimum energy disjoint paths in wireless ad-hoc networks. Wireless Networks, 11(4):401-417, 2005. URL: https://doi.org/10.1007/s11276-005-1765-0.
  47. Hitoshi Suzuki, Takehiro Akama, and Takao Nishizeki. Algorithms for finding internally disjoint paths in a planar graph. Electronics and Communications in Japan (Part III: Fundamental Electronic Science), 72(10):55-67, 1989. URL: https://doi.org/10.1002/ecjc.4430721006.
  48. Hitoshi Suzuki, Takehiro Akama, and Takao Nishizeki. Finding Steiner forests in planar graphs. In Proceedings of the first annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 444-453, 1990. Google Scholar
  49. Carsten Thomassen. 2-linked graphs. European Journal of Combinatorics, 1:371-378, 1980. URL: https://doi.org/10.1016/S0195-6698(80)80039-4.
  50. 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, 2013. URL: https://doi.org/10.1017/CBO9781139506748.005.
  51. Yipu Wang. Max flows in planar graphs with vertex capacities. ACM Trans. Algorithms, 18(1), 2022. URL: https://doi.org/10.1145/3504032.
  52. Marcin Wrochna. Reconfiguration in bounded bandwidth and tree-depth. J. Comput. Syst. Sci., 93:1-10, 2018. URL: https://doi.org/10.1016/j.jcss.2017.11.003.
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