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Shortest Paths in Multimode Graphs

Authors Yael Kirkpatrick , Virginia Vassilevska Williams

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  • Theory of computation → Graph algorithms analysis
Keywords
  • Graph Algorithms
  • Shortest Paths
  • Diameter
  • Radius
  • Fine-Grained Complexity

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Abstract

In this work we study shortest path problems in multimode graphs, a generalization of the min-distance measure introduced by Abboud, Vassilevska W. and Wang in [SODA'16]. A multimode shortest path is the shortest path using one of multiple "modes" of transportation that cannot be combined. This represents real-world scenarios where different modes are not combinable, such as flights operated by different airline alliances. The problem arises naturally in machine learning in the context of learning with multiple embedding. More precisely, a k-multimode graph is a collection of k graphs on the same vertex set and the k-mode distance between two vertices is defined as the minimum among the distances computed in each individual graph. 
We focus on approximating fundamental graph parameters on these graphs, specifically diameter and radius. In undirected multimode graphs we first show an elegant linear time 3-approximation algorithm for 2-mode diameter. We then extend this idea into a general subroutine that can be used as a part of any α-approximation, and use it to construct a 2 and 2.5 approximation algorithm for 2-mode diameter. For undirected radius, we introduce a general scheme that can compute a 3-approximation of the k-mode radius for any k and runs in near linear time in the case of k = O(1). In the directed case we establish an equivalence between approximating 2-mode diameter on DAGs and approximating the min-diameter, while for general graphs we develop novel techniques and provide a linear time algorithm to determine whether the diameter is finite.
We also develop many conditional fine-grained lower bounds for various multimode diameter and radius approximation problems. We are able to show that many of our algorithms are tight under popular fine-grained complexity hypotheses, including our linear time 3-approximation for 3-mode undirected diameter and radius. As part of this effort we propose the first extension to the Hitting Set Hypothesis [SODA'16], which we call the 𝓁-Hitting Set Hypothesis. We use this hypothesis to prove the first parameterized lower bound tradeoff for radius approximation algorithms.

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Yael Kirkpatrick and Virginia Vassilevska Williams. Shortest Paths in Multimode Graphs. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 63:1-63:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.63

Author Details

Yael Kirkpatrick
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Virginia Vassilevska Williams
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Kirkpatrick, Yael: NSF Graduate Research Fellowship under Grant No. 2141064.
  • Vassilevska Williams, Virginia: Supported by NSF Grant CCF-2330048, BSF Grant 2020356 and a Simons Investigator Award.

Acknowledgements

The authors would like to thank Asaf Etgar, Jenny Kaufmann and Surya Mathialagan for helpful discussions.

References

  1. Amir Abboud, Virginia Vassilevska Williams, and Joshua Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '16, pages 377-391, USA, 2016. Society for Industrial and Applied Mathematics. URL: https://doi.org/10.1137/1.9781611974331.CH28.
  2. Arturs Backurs, Liam Roditty, Gilad Segal, Virginia Vassilevska Williams, and Nicole Wein. Toward tight approximation bounds for graph diameter and eccentricities. SIAM Journal on Computing, 50(4):1155-1199, 2021. URL: https://doi.org/10.1137/18M1226737.
  3. Aaron Berger, Jenny Kaufmann, and Virginia Vassilevska Williams. Approximating Min-Diameter: Standard and Bichromatic. In Inge Li Gørtz, Martin Farach-Colton, Simon J. Puglisi, and Grzegorz Herman, editors, 31st Annual European Symposium on Algorithms (ESA 2023), volume 274 of Leibniz International Proceedings in Informatics (LIPIcs), pages 17:1-17:14, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2023.17.
  4. Massimo Cairo, Roberto Grossi, and Romeo Rizzi. New bounds for approximating extremal distances in undirected graphs. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 363-376. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH27.
  5. Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi. On the exact complexity of evaluating quantified k-cnf. In Venkatesh Raman and Saket Saurabh, editors, Parameterized and Exact Computation - 5th International Symposium, IPEC 2010, Chennai, India, December 13-15, 2010. Proceedings, volume 6478 of Lecture Notes in Computer Science, pages 50-59. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-17493-3_7.
  6. Shiri Chechik, Daniel H. Larkin, Liam Roditty, Grant Schoenebeck, Robert Endre Tarjan, and Virginia Vassilevska Williams. Better approximation algorithms for the graph diameter. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1041-1052. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.78.
  7. Shiri Chechik and Tianyi Zhang. Constant approximation of min-distances in near-linear time. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 896-906. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00089.
  8. Lenore Cowen and Christopher G. Wagner. Compact roundtrip routing for digraphs. In Robert Endre Tarjan and Tandy J. Warnow, editors, Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, 17-19 January 1999, Baltimore, Maryland, USA, pages 885-886. ACM/SIAM, 1999. URL: http://dl.acm.org/citation.cfm?id=314500.315068.
  9. Mina Dalirrooyfard and Jenny Kaufmann. Approximation Algorithms for Min-Distance Problems in DAGs. 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 60:1-60:17, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.60.
  10. Mina Dalirrooyfard, Ray Li, and Virginia Vassilevska Williams. Hardness of approximate diameter: Now for undirected graphs. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 1021-1032. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00102.
  11. Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, and Nicole Wein. Tight approximation algorithms for bichromatic graph diameter and related problems. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 47:1-47:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPICS.ICALP.2019.47.
  12. Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, Nicole Wein, Yinzhan Xu, and Yuancheng Yu. Approximation Algorithms for Min-Distance Problems. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), volume 132 of Leibniz International Proceedings in Informatics (LIPIcs), pages 46:1-46:14, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.46.
  13. Mina Dalirrooyfard and Nicole Wein. Tight conditional lower bounds for approximating diameter in directed graphs. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1697-1710. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451130.
  14. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  15. Lung-Hao Lee and Yi Lu. Multiple embeddings enhanced multi-graph neural networks for chinese healthcare named entity recognition. IEEE Journal of Biomedical and Health Informatics, 25(7):2801-2810, 2021. URL: https://doi.org/10.1109/JBHI.2020.3048700.
  16. Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 515-524. ACM, 2013. URL: https://doi.org/10.1145/2488608.2488673.
  17. Liam Roditty and Uri Zwick. On dynamic shortest paths problems. In Algorithms-ESA 2004: 12th Annual European Symposium, Bergen, Norway, September 14-17, 2004. Proceedings 12, pages 580-591. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-30140-0_52.
  18. Fei Tian, Hanjun Dai, Jiang Bian, Bin Gao, Rui Zhang, Enhong Chen, and Tie-Yan Liu. A probabilistic model for learning multi-prototype word embeddings. In Proceedings of COLING 2014, the 25th International Conference on Computational Linguistics: Technical Papers, pages 151-160, 2014. URL: https://aclanthology.org/C14-1016/.
  19. Virginia Vassilevska Williams. On some fine-grained questions in algorithms and complexity. In Proceedings of the ICM, volume 3, pages 3431-3472. World Scientific, 2018. Google Scholar
  20. Zixuan Xu Virginia Vassilevska Williams, Yinzhan Xu and Refei Zhou. New bounds for matrix multiplication: from alpha to omega. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3792-3835, 2024. URL: https://doi.org/10.1137/1.9781611977912.134.
  21. Canwen Xu, Feiyang Wang, Jialong Han, and Chenliang Li. Exploiting multiple embeddings for chinese named entity recognition. In Proceedings of the 28th ACM International Conference on Information and Knowledge Management, CIKM '19, pages 2269-2272, New York, NY, USA, 2019. Association for Computing Machinery. URL: https://doi.org/10.1145/3357384.3358117.
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