Document Open Access Logo

Two-Sets Cut-Uncut on Planar Graphs

Authors Matthias Bentert, Pål Grønås Drange , Fedor V. Fomin , Petr A. Golovach , Tuukka Korhonen

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.22.pdf
  • Filesize: 0.88 MB
  • 18 pages

Document Identifiers

Author Details

Matthias Bentert
  • University of Bergen, Norway
Pål Grønås Drange
  • University of Bergen, Norway
Fedor V. Fomin
  • University of Bergen, Norway
Petr A. Golovach
  • University of Bergen, Norway
Tuukka Korhonen
  • University of Bergen, Norway

Funding

The research leading to these results has received funding from the Research Council of Norway via the project BWCA (grant no. 314528) and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416).

Cite As Get BibTex

Matthias Bentert, Pål Grønås Drange, Fedor V. Fomin, Petr A. Golovach, and Tuukka Korhonen. Two-Sets Cut-Uncut on Planar Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.22

Abstract

We study Two-Sets Cut-Uncut on planar graphs. Therein, one is given an undirected planar graph G and two disjoint sets S and T of vertices as input. The question is, what is the minimum number of edges to remove from G, such that all vertices in S are separated from all vertices in T, while maintaining that every vertex in S, and respectively in T, stays in the same connected component. We show that this problem can be solved in 2^{|S|+|T|} n^𝒪(1) time with a one-sided-error randomized algorithm. Our algorithm implies a polynomial-time algorithm for the network diversion problem on planar graphs, which resolves an open question from the literature. More generally, we show that Two-Sets Cut-Uncut is fixed-parameter tractable when parameterized by the number r of faces in a planar embedding covering the terminals S ∪ T, by providing a 2^𝒪(r) n^𝒪(1)-time algorithm.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • planar graphs
  • cut-uncut
  • group-constrained paths

Metrics

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

Related Versions

References

  1. Geir Agnarsson and Raymond Greenlaw. Graph theory: Modeling, applications, and algorithms. Pearson, 2006. Google Scholar
  2. Matthias Bentert, Pål Grønås Drange, Fedor V. Fomin, Petr A. Golovach, and Tuukka Korhonen. Two-sets cut-uncut on planar graphs, 2023. URL: https://arxiv.org/abs/2305.01314.
  3. Marshall Bern. Faster exact algorithms for steiner trees in planar networks. Networks, 20(1):109-120, 1990. Google Scholar
  4. Ivona Bezáková and Zachary Langley. Minimum planar multi-sink cuts with connectivity priors. In Proceedings of the 39th International Symposium on Mathematical Foundations of Computer Science (MFCS), pages 94-105. Springer, 2014. Google Scholar
  5. Daniel Bienstock and Clyde L. Monma. On the complexity of covering vertices by faces in a planar graph. SIAM Journal on Computing, 17(1):53-76, 1988. Google Scholar
  6. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Narrow sieves for parameterized paths and packings. Journal of Computer and System Sciences, 87:119-139, 2017. Google Scholar
  7. Andreas Björklund, Thore Husfeldt, and Nina Taslaman. Shortest cycle through specified elements. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1747-1753. Society for Industrial and Applied Mathematics, 2012. Google Scholar
  8. Rajesh Chitnis, Marek Cygan, MohammadTaghi Hajiaghayi, Marcin Pilipczuk, and Michal Pilipczuk. Designing FPT algorithms for cut problems using randomized contractions. SIAM Journal on Computing, 45(4):1171-1229, 2016. Google Scholar
  9. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms. MIT Press, 2009. Google Scholar
  10. Christopher A. Cullenbine, R. Kevin Wood, and Alexandra M. Newman. Theoretical and computational advances for network diversion. Networks, 62(3):225-242, 2013. Google Scholar
  11. Norman D. Curet. The network diversion problem. Military Operations Research, 6(2):35-44, 2001. Google Scholar
  12. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  13. Marek Cygan, Pawel Komosa, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, Saket Saurabh, and Magnus Wahlström. Randomized contractions meet lean decompositions. ACM Transactions on Algorithms, 17(1):6:1-6:30, 2021. Google Scholar
  14. Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, and Jakub Onufry Wojtaszczyk. Solving the 2-disjoint connected subgraphs problem faster than 2ⁿ. Algorithmica, 70(2):195-207, 2014. Google Scholar
  15. Elias Dahlhaus, David S. Johnson, Christos H. Papadimitriou, Paul D. Seymour, and Mihalis Yannakakis. The complexity of multiterminal cuts. SIAM Journal on Computing, 23(4):864-894, 1994. Google Scholar
  16. Ulrich Derigs. An efficient Dijkstra-like labeling method for computing shortest odd/even paths. Information Processing Letters, 21(5):253-258, 1985. Google Scholar
  17. Reinhard Diestel. Graph Theory. Springer, 2012. Google Scholar
  18. Qi Duan, Haadi Jafarian, Ehab Al-Shaer, and Jinhui Xu. On DDoS attack related minimum cut problems. CoRR, abs/1412.3359, 2015. URL: https://arxiv.org/abs/1412.3359.
  19. Qi Duan and Jinhui Xu. On the connectivity preserving minimum cut problem. Journal of Computer and System Sciences, 80(4):837-848, 2014. Google Scholar
  20. Eduard Eiben, Tomohiro Koana, and Magnus Wahlström. Determinantal sieving. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 377-423. Society for Industrial and Applied Mathematics, 2024. Google Scholar
  21. Ranel E. Erickson, Clyde L. Monma, and Arthur F. Jr. Veinott. Send-and-split method for minimum-concave-cost network flows. Mathematics of Operations Research, 12(4):634-664, 1987. Google Scholar
  22. Ozgur Erken. A branch-and-bound algorithm for the network diversion problem. Master’s thesis, Naval Postgraduate School, 2002. Google Scholar
  23. Arnold Filtser. A face cover perspective to 𝓁₁ embeddings of planar graphs. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1945-1954. Society for Industrial and Applied Mathematics, 2020. Google Scholar
  24. Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, Kirill Simonov, and Giannos Stamoulis. Fixed-parameter tractability of maximum colored path and beyond. In Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3700-3712. SIAM, 2023. Google Scholar
  25. Chris Gray, Frank Kammer, Maarten Löffler, and Rodrigo I. Silveira. Removing local extrema from imprecise terrains. Computational Geometry, 45(7):334-349, 2012. Google Scholar
  26. Martin Grötschel and William R. Pulleyblank. Weakly bipartite graphs and the max-cut problem. Operations Research Letters, 1(1):23-27, 1981. Google Scholar
  27. John E. Hopcroft and Robert E. Tarjan. Efficient planarity testing. Journal of the ACM, 21(4):549-568, 1974. Google Scholar
  28. Yoichi Iwata and Yutaro Yamaguchi. Finding a shortest non-zero path in group-labeled graphs. Combinatorica, 42(S2):1253-1282, 2022. Google Scholar
  29. Benjamin S. Kallemyn. Modeling Network Interdiction Tasks. Doctoral thesis, Air Force Institute Of Technology, 2015. Google Scholar
  30. Frank Kammer and Torsten Tholey. The complexity of minimum convex coloring. Discrete Applied Mathematics, 160(6):810-833, 2012. Google Scholar
  31. Sándor Kisfaludi-Bak, Jesper Nederlof, and Erik Jan van Leeuwen. Nearly ETH-tight algorithms for planar Steiner tree with terminals on few faces. ACM Transactions on Algorithms, 16(3):1-30, 2020. Google Scholar
  32. Philip N. Klein and Dániel Marx. Solving planar k-terminal cut in O(n^c√k) time. In Proceedings of the 39th International Colloquium on Automata, Languages, and Programming (ICALP), pages 569-580. Springer, 2012. Google Scholar
  33. Yusuke Kobayashi and Sho Toyooka. Finding a shortest non-zero path in group-labeled graphs via permanent computation. Algorithmica, 77(4):1128-1142, 2017. Google Scholar
  34. Ioannis Koutis. Faster algebraic algorithms for path and packing problems. In Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP), pages 575-586. Springer, 2008. Google Scholar
  35. Robert Krauthgamer, James R. Lee, and Havana I. Rika. Flow-cut gaps and face covers in planar graphs. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 525-534. SIAM, 2019. Google Scholar
  36. Chungmok Lee, Donghyun Cho, and Sungsoo Park. A combinatorial benders decomposition algorithm for the directed multiflow network diversion problem. Military Operations Research, 24(1):23-40, 2019. Google Scholar
  37. Dániel Marx. A tight lower bound for planar multiway cut with fixed number of terminals. In Proceedings of the 39th International Colloquium on Automata, Languages, and Programming (ICALP), pages 677-688. Springer, 2012. Google Scholar
  38. Luke Mathieson and Stefan Szeider. Editing graphs to satisfy degree constraints: A parameterized approach. Journal of Computer and System Sciences, 78(1):179-191, 2012. Google Scholar
  39. Sukanya Pandey and Erik Jan van Leeuwen. Planar multiway cut with terminals on few faces. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2032-2063. Society for Industrial and Applied Mathematics, 2022. Google Scholar
  40. Daniël Paulusma and Johan M. M. van Rooij. On partitioning a graph into two connected subgraphs. Theor. Comput. Sci., 412(48):6761-6769, 2011. URL: https://doi.org/10.1016/j.tcs.2011.09.001.
  41. Ashutosh Rai, M. S. Ramanujan, and Saket Saurabh. A parameterized algorithm for mixed-cut. In Proceedings of the 12th Latin American Symposium (LATIN), pages 672-685. Springer, 2016. Google Scholar
  42. Jacob T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM, 27(4):701-717, 1980. Google Scholar
  43. Jan Arne Telle and Yngve Villanger. Connecting terminals and 2-disjoint connected subgraphs. In Proceedings of the 39th International Workshop on Graph-Theoretic Concepts in Computer Science (WG), pages 418-428. Springer, 2013. Google Scholar
  44. Pim van't Hof, Daniël Paulusma, and Gerhard J. Woeginger. Partitioning graphs into connected parts. Theoretical Computer Science, 410(47-49):4834-4843, 2009. Google Scholar
  45. Ryan Williams. Finding paths of length k in O^*(2^k) time. Information Processing Letters, 109(6):315-318, 2009. Google Scholar
  46. Richard Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposiumon Symbolic and Algebraic Computation (EUROSAM), pages 216-226. Springer, 1979. 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact