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(Almost-)Optimal FPT Algorithm and Kernel for T-Cycle on Planar Graphs

Authors Harmender Gahlawat , Abhishek Rathod , Meirav Zehavi

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Graph algorithms
Keywords
  • FPT Algorithms
  • Kernelization
  • T-Cycle
  • Subexponential Algorithmms

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Abstract

Research of cycles through specific vertices is a central topic in graph theory. In this context, we focus on a well-studied computational problem, T-Cycle: given an undirected n-vertex graph G and a set of k vertices T ⊆ V(G) termed terminals, the objective is to determine whether G contains a simple cycle C through all the terminals. Our contribution is twofold: (i) We provide a 2^{O(√klog k)}⋅ n-time fixed-parameter deterministic algorithm for T-Cycle on planar graphs; (ii) We provide a k^{O(1)}⋅ n-time deterministic kernelization algorithm for T-Cycle on planar graphs where the produced instance is of size klog^{O(1)}k. 
Both of our algorithms are optimal in terms of both k and n up to (poly)logarithmic factors in k under the ETH. In fact, our algorithms are the first subexponential-time fixed-parameter algorithm for T-Cycle on planar graphs, as well as the first polynomial kernel for T-Cycle on planar graphs. This substantially improves upon/expands the known literature on the parameterized complexity of the problem.

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Harmender Gahlawat, Abhishek Rathod, and Meirav Zehavi. (Almost-)Optimal FPT Algorithm and Kernel for T-Cycle on Planar Graphs. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 82:1-82:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.82

Author Details

Harmender Gahlawat
  • Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Étienne, LIMOS, 63000 Clermont-Ferrand, France
  • G-SCOP, Grenoble-INP, Grenoble, France
Abhishek Rathod
  • Ben-Gurion University of the Negev, Beersheba, Israel
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beersheba, Israel

Funding

This research was supported by the European Research Council (ERC) grant titled PARAPATH (101039913) and by the ISF grant with number ISF-1470/24 and the IDEX-ISITE initiative CAP 20-25 (ANR-16-IDEX-0001), the International Research Center "Innovation Transportation and Production Systems" of the I-SITE CAP 20-25, and the ANR project GRALMECO (ANR-21-CE48-0004).

References

  1. Isolde Adler, Stavros G Kolliopoulos, Philipp Klaus Krause, Daniel Lokshtanov, Saket Saurabh, and Dimitrios Thilikos. Tight bounds for linkages in planar graphs. In Automata, Languages and Programming: 38th International Colloquium, ICALP 2011, Zurich, Switzerland, July 4-8, 2011, Proceedings, Part I 38, pages 110-121. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22006-7_10.
  2. 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.
  3. Andreas Björklund. Determinant sums for undirected hamiltonicity. SIAM J. Comput., 43(1):280-299, 2014. URL: https://doi.org/10.1137/110839229.
  4. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Narrow sieves for parameterized paths and packings. J. Comput. Syst. Sci., 87:119-139, 2017. URL: https://doi.org/10.1016/J.JCSS.2017.03.003.
  5. Andreas Björklund, Thore Husfeldt, and Nina Taslaman. Shortest cycle through specified elements. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1747-1753, 2012. URL: https://doi.org/10.1137/1.9781611973099.139.
  6. Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423-434, 2009. URL: https://doi.org/10.1016/J.JCSS.2009.04.001.
  7. Hans L Bodlaender, Fedor V Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M Thilikos. (meta) kernelization. Journal of the ACM (JACM), 63(5):1-69, 2016. URL: https://doi.org/10.1145/2973749.
  8. John Adrian Bondy and László Lovász. Cycles through specified vertices of a graph. Combinatorica, 1:117-140, 1981. URL: https://doi.org/10.1007/BF02579268.
  9. Liming Cai, Jianer Chen, Rodney G Downey, and Michael R Fellows. Advice classes of parameterized tractability. Annals of pure and applied logic, 84(1):119-138, 1997. URL: https://doi.org/10.1016/S0168-0072(95)00020-8.
  10. Chandra Chekuri and Julia Chuzhoy. Polynomial bounds for the grid-minor theorem. J. ACM, 63(5):40:1-40:65, 2016. URL: https://doi.org/10.1145/2820609.
  11. Kyungjin Cho, Eunjin Oh, and Seunghyeok Oh. Parameterized algorithm for the disjoint path problem on planar graphs: Exponential in k² and linear in n. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3734-3758. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH144.
  12. 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.
  13. Gabriel Andrew Dirac. In abstrakten graphen vorhandene vollständige 4-graphen und ihre unterteilungen. Mathematische Nachrichten, 22(1-2):61-85, 1960. Google Scholar
  14. Frederic Dorn, Fedor V. Fomin, and Dimitrios M. Thilikos. Catalan structures and dynamic programming in h-minor-free graphs. J. Comput. Syst. Sci., 78(5):1606-1622, 2012. URL: https://doi.org/10.1016/J.JCSS.2012.02.004.
  15. Péter L Erdős and Ervin Győri. Any four independent edges of a 4-connected graph are contained in a circuit. Acta Mathematica Hungarica, 46:311-313, 1985. Google Scholar
  16. Michael R. Fellows. The lost continent of polynomial time: Preprocessing and kernelization. In Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings, pages 276-277, 2006. URL: https://doi.org/10.1007/11847250_25.
  17. Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: theory of parameterized preprocessing. Cambridge University Press, 2019. Google Scholar
  18. 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.
  19. Harmender Gahlawat, Abhishek Rathod, and Meirav Zehavi. (almost-)optimal fpt algorithm and kernel for t-cycle on planar graphs, 2025. URL: https://arxiv.org/abs/2504.19301.
  20. Roland Häggkvist and Carsten Thomassen. Circuits through specified edges. Discrete mathematics, 41(1):29-34, 1982. URL: https://doi.org/10.1016/0012-365X(82)90078-4.
  21. Derek A. Holton, Brendan D. McKay, Michael D. Plummer, and Carsten Thomassen. A nine point theorem for 3-connected graphs. Combinatorica, 2:53-62, 1982. URL: https://doi.org/10.1007/BF02579281.
  22. Ken-ichi Kawarabayashi. One or two disjoint circuits cover independent edges: Lovász-woodall conjecture. Journal of Combinatorial Theory, Series B, 84(1):1-44, 2002. URL: https://doi.org/10.1006/JCTB.2001.2059.
  23. Ken-ichi Kawarabayashi. Cycles through a prescribed vertex set in n-connected graphs. Journal of Combinatorial Theory, Series B, 90(2):315-323, 2004. URL: https://doi.org/10.1016/J.JCTB.2003.08.002.
  24. Ken-ichi Kawarabayashi. An improved algorithm for finding cycles through elements. In Integer Programming and Combinatorial Optimization: 13th International Conference, IPCO 2008 Bertinoro, Italy, May 26-28, 2008 Proceedings 13, pages 374-384. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-68891-4_26.
  25. Ken-ichi Kawarabayashi, Yusuke Kobayashi, and Bruce A. Reed. The disjoint paths problem in quadratic time. J. Comb. Theory B, 102(2):424-435, 2012. URL: https://doi.org/10.1016/J.JCTB.2011.07.004.
  26. Ken-ichi Kawarabayashi, Zhentao Li, and Bruce Reed. Recognizing a totally odd k 4-subdivision, parity 2-disjoint rooted paths and a parity cycle through specified elements. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 318-328. SIAM, 2010. URL: https://doi.org/10.1137/1.9781611973075.27.
  27. Yusuke Kobayashi and Ken-ichi Kawarabayashi. Algorithms for finding an induced cycle in planar graphs and bounded genus graphs. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1146-1155. SIAM, 2009. URL: https://doi.org/10.1137/1.9781611973068.124.
  28. Tuukka Korhonen, Michał Pilipczuk, and Giannos Stamoulis. Minor containment and disjoint paths in almost-linear time. FOCS, 2024. Google Scholar
  29. Andrea S. LaPaugh and Ronald L. Rivest. The subgraph homeomorphism problem. J. Comput. Syst. Sci., 20(2):133-149, 1980. URL: https://doi.org/10.1016/0022-0000(80)90057-4.
  30. Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Efficient graph minors theory and parameterized algorithms for (planar) disjoint paths. In Treewidth, Kernels, and Algorithms - Essays Dedicated to Hans L. Bodlaender on the Occasion of His 60th Birthday, pages 112-128, 2020. URL: https://doi.org/10.1007/978-3-030-42071-0_9.
  31. Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Efficient computation of representative weight functions with applications to parameterized counting (extended version). In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 179-198, 2021. URL: https://doi.org/10.1137/1.9781611976465.13.
  32. Dániel Marx, Marcin Pilipczuk, and Michal Pilipczuk. On subexponential parameterized algorithms for steiner tree and directed subset TSP on planar graphs. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 474-484, 2018. URL: https://doi.org/10.1109/FOCS.2018.00052.
  33. Bruce Reed. Rooted routing in the plane. Discrete Applied Mathematics, 57(2-3):213-227, 1995. URL: https://doi.org/10.1016/0166-218X(94)00104-L.
  34. Neil Robertson and Paul D. Seymour. Graph minors .xiii. the disjoint paths problem. J. Comb. Theory B, 63(1):65-110, 1995. URL: https://doi.org/10.1006/JCTB.1995.1006.
  35. Daniel P Sanders. On circuits through five edges. Discrete Mathematics, 159(1-3):199-215, 1996. URL: https://doi.org/10.1016/0012-365X(95)00079-C.
  36. Carsten Thomassen. Note on circuits containing specified edges. Journal of Combinatorial Theory, Series B, 22(3):279-280, 1977. URL: https://doi.org/10.1016/0095-8956(77)90073-9.
  37. Dekel Tsur. Faster deterministic parameterized algorithm for k-path. Theor. Comput. Sci., 790:96-104, 2019. URL: https://doi.org/10.1016/J.TCS.2019.04.024.
  38. Magnus Wahlström. Abusing the tutte matrix: An algebraic instance compression for the k-set-cycle problem. In 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, February 27 - March 2, 2013, Kiel, Germany, pages 341-352, 2013. URL: https://doi.org/10.4230/LIPICS.STACS.2013.341.
  39. Michal Wlodarczyk and Meirav Zehavi. Planar disjoint paths, treewidth, and kernels. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 649-662, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00044.
  40. Douglas R Woodall. Circuits containing specified edges. J. Comb. Theory, Ser. B, 22(3):274-278, 1977. URL: https://doi.org/10.1016/0095-8956(77)90072-7.
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