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Detecting Disjoint Shortest Paths in Linear Time and More

Authors Shyan Akmal , Virginia Vassilevska Williams, Nicole Wein

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Author Details

Shyan Akmal
  • MIT, EECS and CSAIL, Cambridge, MA, USA
Virginia Vassilevska Williams
  • MIT, EECS and CSAIL, Cambridge, MA, USA
Nicole Wein
  • University of Michigan, Ann Arbor, MI, USA

Funding

  • Akmal, Shyan: Supported by Virginia Vassilevska Williams' Simons Investigator Award.
  • Vassilevska Williams, Virginia: Supported by NSF Grant CCF-2330048, BSF Grant 2020356, and a Simons Investigator Award.

Acknowledgements

We thank an anonymous reviewer for pointing out issues with a proof in a previous version of this work, and another anonymous reviewer for simplifying our construction of covering families. We thank Malte Renken and André Nichterlein for answering questions about previous work, as well as Matthias Bentert for telling us about his and his coauthors' work [Matthias Bentert et al., 2024], which independently proves Theorem 6 from this paper. The first author additionally thanks Ryan Williams and Zixuan Xu for conversations about early versions of ideas presented here, and Yinzhan Xu for nice discussions about the arguments in this paper.

Cite As Get BibTex

Shyan Akmal, Virginia Vassilevska Williams, and Nicole Wein. Detecting Disjoint Shortest Paths in Linear Time and More. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.9

Abstract

In the k-Disjoint Shortest Paths (k-DSP) problem, we are given a graph G (with positive edge weights) on n nodes and m edges with specified source vertices s_1, … , s_k, and target vertices t_1, … , t_k, and are tasked with determining if G contains vertex-disjoint (s_i,t_i)-shortest paths. For any constant k, it is known that k-DSP can be solved in polynomial time over undirected graphs and directed acyclic graphs (DAGs). However, the exact time complexity of k-DSP remains mysterious, with large gaps between the fastest known algorithms and best conditional lower bounds. In this paper, we obtain faster algorithms for important cases of k-DSP, and present better conditional lower bounds for k-DSP and its variants.
Previous work solved 2-DSP over weighted undirected graphs in O(n⁷) time, and weighted DAGs in O(mn) time. For the main result of this paper, we present optimal linear time algorithms for solving 2-DSP on weighted undirected graphs and DAGs. Our linear time algorithms are algebraic however, and so only solve the detection rather than search version of 2-DSP (we show how to solve the search version in O(mn) time, which is faster than the previous best runtime in weighted undirected graphs, but only matches the previous best runtime for DAGs).
We also obtain a faster algorithm for k-Edge Disjoint Shortest Paths (k-EDSP) in DAGs, the variant of k-DSP where one seeks edge-disjoint instead of vertex-disjoint paths between sources and their corresponding targets. Algorithms for k-EDSP on DAGs from previous work take Ω(m^k) time. We show that k-EDSP can be solved over DAGs in O(mn^{k-1}) time, matching the fastest known runtime for solving k-DSP over DAGs.
Previous work established conditional lower bounds for solving k-DSP and its variants via reductions from detecting cliques in graphs. Prior work implied that k-Clique can be reduced to 2k-DSP in DAGs and undirected graphs with O((kn)²) nodes. We improve this reduction, by showing how to reduce from k-Clique to k-DSP in DAGs and undirected graphs with O((kn)²) nodes (halving the number of paths needed in the reduced instance). A variant of k-DSP is the k-Disjoint Paths (k-DP) problem, where the solution paths no longer need to be shortest paths. Previous work reduced from k-Clique to p-DP in DAGs with O(kn) nodes, for p = k + k(k-1)/2. We improve this by showing a reduction from k-Clique to p-DP, for p = k + ⌊k²/4⌋.
Under the k-Clique Hypothesis from fine-grained complexity, our results establish better conditional lower bounds for k-DSP for all k ≥ 4, and better conditional lower bounds for p-DP for all p ≤ 4031. Notably, our work gives the first nontrivial conditional lower bounds 4-DP in DAGs and 4-DSP in undirected graphs and DAGs. Before our work, nontrivial conditional lower bounds were only known for k-DP and k-DSP on such graphs when k ≥ 6.

Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
Keywords
  • disjoint shortest paths
  • algebraic graph algorithms
  • disjoint paths
  • fine-grained complexity
  • clique

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References

  1. Maxim Akhmedov. Faster 2-disjoint-shortest-paths algorithm. In Computer Science endash Theory and Applications, pages 103-116. Springer International Publishing, 2020. URL: https://doi.org/10.1007/978-3-030-50026-9_7.
  2. Shyan Akmal, Virginia Vassilevska Williams, and Nicole Wein. Detecting Disjoint Shortest Paths in Linear Time and More, 2024. URL: https://arxiv.org/abs/2404.15916.
  3. Matthias Bentert, Fedor V. Fomin, and Petr A. Golovach. Tight approximation and kernelization bounds for vertex-disjoint shortest paths, 2024. URL: https://arxiv.org/abs/2402.15348.
  4. Matthias Bentert, André Nichterlein, Malte Renken, and Philipp Zschoche. Using a Geometric Lens to Find k Disjoint Shortest Paths. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), volume 198 of Leibniz International Proceedings in Informatics (LIPIcs), pages 26:1-26:14, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.26.
  5. Matthias Bentert, André Nichterlein, Malte Renken, and Philipp Zschoche. Using a geometric lens to find k disjoint shortest paths, 2020. URL: https://doi.org/10.48550/arXiv.2007.12502.
  6. Kristof Berczi and Yusuke Kobayashi. The Directed Disjoint Shortest Paths Problem. In Kirk Pruhs and Christian Sohler, editors, 25th Annual European Symposium on Algorithms (ESA 2017), volume 87 of Leibniz International Proceedings in Informatics (LIPIcs), pages 13:1-13:13, Dagstuhl, Germany, 2017. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2017.13.
  7. Andreas Björklund and Thore Husfeldt. Shortest two disjoint paths in polynomial time. SIAM Journal on Computing, 48(6):1698-1710, January 2019. URL: https://doi.org/10.1137/18m1223034.
  8. Andreas Björklund, Thore Husfeldt, and Petteri Kaski. The shortest even cycle problem is tractable. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. ACM, June 2022. URL: https://doi.org/10.1145/3519935.3520030.
  9. 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. Society for Industrial and Applied Mathematics, January 2012. URL: https://doi.org/10.1137/1.9781611973099.139.
  10. Rajesh Chitnis. A tight lower bound for edge-disjoint paths on planar dags, 2021. URL: https://doi.org/10.48550/arXiv.2101.10742.
  11. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer International Publishing, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  12. Marek Cygan, Harold N. Gabow, and Piotr Sankowski. Algorithmic applications of baur-strassen’s theorem. Journal of the ACM, 62(4):1-30, September 2015. URL: https://doi.org/10.1145/2736283.
  13. Mina Dalirrooyfard and Virginia Vassilevska Williams. Induced cycles and paths are harder than you think. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 531-542. IEEE, 2022. Google Scholar
  14. Eduard Eiben, Tomohiro Koana, and Magnus Wahlström. Determinantal sieving, 2023. URL: https://arxiv.org/abs/2304.02091.
  15. Tali Eilam-Tzoreff. The disjoint shortest paths problem. Discrete Applied Mathematics, 85(2):113-138, June 1998. URL: https://doi.org/10.1016/s0166-218x(97)00121-2.
  16. Friedrich Eisenbrand and Fabrizio Grandoni. On the complexity of fixed parameter clique and dominating set. Theoretical Computer Science, 326(1-3):57-67, October 2004. URL: https://doi.org/10.1016/j.tcs.2004.05.009.
  17. Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, Kirill Simonov, and Giannos Stamoulis. Fixed-parameter tractability of maximum colored path and beyond, 2022. URL: https://arxiv.org/abs/2207.07449.
  18. Steven Fortune, John Hopcroft, and James Wyllie. The directed subgraph homeomorphism problem. Theoretical Computer Science, 10(2):111-121, February 1980. URL: https://doi.org/10.1016/0304-3975(80)90009-2.
  19. Ken-ichi Kawarabayashi, Yusuke Kobayashi, and Bruce Reed. The disjoint paths problem in quadratic time. Journal of Combinatorial Theory, Series B, 102(2):424-435, March 2012. URL: https://doi.org/10.1016/j.jctb.2011.07.004.
  20. Tuukka Korhonen, Michał Pilipczuk, and Giannos Stamoulis. Minor containment and disjoint paths in almost-linear time, 2024. URL: https://arxiv.org/abs/2404.03958.
  21. Willian Lochet. A polynomial time algorithm for the k-disjoint shortest paths problem. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 169-178. Society for Industrial and Applied Mathematics, January 2021. URL: https://doi.org/10.1137/1.9781611976465.12.
  22. N. Robertson and P.D. Seymour. Graph minors .XIII. the disjoint paths problem. Journal of Combinatorial Theory, Series B, 63(1):65-110, January 1995. URL: https://doi.org/10.1006/jctb.1995.1006.
  23. Aleksandrs Slivkins. Parameterized tractability of edge-disjoint paths on directed acyclic graphs. SIAM Journal on Discrete Mathematics, 24(1):146-157, January 2010. URL: https://doi.org/10.1137/070697781.
  24. Torsten Tholey. Solving the 2-disjoint paths problem in nearly linear time. Theory of Computing Systems, 39(1):51-78, November 2005. URL: https://doi.org/10.1007/s00224-005-1256-9.
  25. Torsten Tholey. Linear time algorithms for two disjoint paths problems on directed acyclic graphs. Theoretical Computer Science, 465:35-48, December 2012. URL: https://doi.org/10.1016/j.tcs.2012.09.025.
  26. M. Thorup. Undirected single source shortest paths in linear time. In Proceedings 38th Annual Symposium on Foundations of Computer Science. IEEE Comput. Soc, 1997. URL: https://doi.org/10.1109/sfcs.1997.646088.
  27. Virginia Vassilevska Williams. On some fine-grained questions in algorithms and complexity. In Proceedings of the international congress of mathematicians: Rio de janeiro 2018, pages 3447-3487. World Scientific, 2018. Google Scholar
  28. Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. New bounds for matrix multiplication: from alpha to omega, 2023. URL: https://arxiv.org/abs/2307.07970.
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