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Additive, Near-Additive, and Multiplicative Approximations for APSP in Weighted Undirected Graphs: Trade-Offs and Algorithms

Authors Liam Roditty , Ariel Sapir

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Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
Keywords
  • Graph
  • Shortest Paths
  • Weighted Graphs
  • Approximation
  • Undirected
  • Single Source Shortest-Paths
  • Multi-Source Shortest-Paths
  • All-Pairs Shortest-Paths
  • SSSP
  • MSSP
  • MSASP
  • APSP
  • APASP

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Abstract

We present a +2∑_{i=1}^{k+1} W_i-APASP algorithm for dense weighted graphs with a runtime of Õ(n^{2+1/(3k+2)), where W_i is the weight of an i^th heaviest edge on a shortest path between two vertices. Dor, Halperin and Zwick [FOCS'96 and SICOMP'00] introduced two algorithms for the commensurate unweighted +2⋅ (k+1)-APASP problem: one for sparse graphs with a runtime of Õ(n^{2-1/(k+2)} m^{1/(k+2)}) and one for dense graphs with a runtime of Õ(n^{2+1/(3k+2)}). Subsequently, Cohen and Zwick [SODA'97 and JALG'01] adapted the algorithm for sparse graphs to the weighted setting, namely a +2∑_{i=1}^{k+1} W_i-APASP algorithm with the same Õ(n^{2-1/(k+2)} m^{1/(k+2)}) runtime. We fill the nearly three decades old gap by providing an algorithm for dense weighted graphs, matching the runtime for the unweighted setting. 
In addition, we explore nearly additive APASP, where the multiplicative stretch is 1+ε. We present a (1+ε, min{2W₁,4W₂})-APASP algorithm with a runtime of Õ((1/ε)^{O(1)} ⋅ n^{2.15135313} ⋅ log W). This improves upon Saha and Ye [SODA'24], which had the same runtime, yet (1+ε, 2W₁)-APASP. 
For pure multiplicative APASP, we present a (7/3+ε)-APASP algorithm with a runtime of Õ((1/ε)^{O(1)} ⋅ n^{2.15135313} ⋅ log W). This improves, for dense graphs, the Õ(nm^{2/3}+n²) runtime of the 7/3-APASP algorithm by Baswana and Kavitha [FOCS'06 and SICOMP'10], at the cost of introducing an additional ε to the multiplicative stretch. 
We further view this result in a broader framework of ((3𝓁+4)/(𝓁+2) + ε)-APASP algorithms, similarly to the family of (3𝓁+4)/(𝓁+2)-APASP algorithms by Akav and Roditty [ESA'21]. This also generalizes the (2+ε)-APASP algorithm by Dory, Forster, Kirkpatrick, Nazari, Vassilevska Williams, and de Vos [SODA'24].
Finally, we show that it is possible to "bypass" an Ω̃ (n^ω) conditional lower bound by Dor, Halperin, and Zwick for α-APASP with α < 2, by allowing an additive component to the approximation (e.g. a ((6k+3)/(3k+2),∑_{i=1}^{k+1} W_i)-APASP with Õ(n^{2+1/(3k+2)}) runtime.).

Cite As Get BibTex

Liam Roditty and Ariel Sapir. Additive, Near-Additive, and Multiplicative Approximations for APSP in Weighted Undirected Graphs: Trade-Offs and Algorithms. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 50:1-50:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FSTTCS.2025.50

Author Details

Liam Roditty
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
Ariel Sapir
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel

References

  1. Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Stephen Kobourov, and Richard Spence. Weighted additive spanners. arXiv, 2021. URL: https://arxiv.org/abs/2002.07152.
  2. Donald Aingworth, Chandra Chekuri, Piotr Indyk, and Rajeev Motwani. Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM Journal on Computing, 28(4):1167-1181, 1999. URL: https://doi.org/10.1137/S0097539796303421.
  3. Maor Akav and Liam Roditty. A unified approach for all pairs approximate shortest paths in weighted undirected graphs. In 29th Annual European Symposium on Algorithms (ESA 2021), volume 204 of Leibniz International Proceedings in Informatics (LIPIcs), pages 4:1-4:18, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.4.
  4. Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. More asymmetry yields faster matrix multiplication. arXiv, abs/2404.16349, 2024. URL: https://doi.org/10.48550/arXiv.2404.16349.
  5. Andris Ambainis, Yuval Filmus, and François Le Gall. Fast matrix multiplication: Limitations of the coppersmith-winograd method. Proceedings of the forty-seventh annual ACM symposium on Theory of Computing, 2014. URL: https://api.semanticscholar.org/CorpusID:8332797.
  6. Surender Baswana and Telikepalli Kavitha. Faster algorithms for all-pairs approximate shortest paths in undirected graphs. SIAM Journal on Computing, 39(7):2865-2896, 2010. URL: https://doi.org/10.1137/080737174.
  7. Piotr Berman and Shiva Prasad Kasiviswanathan. Faster approximation of distances in graphs. In Algorithms and Data Structures, pages 541-552, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-540-73951-7_47.
  8. Karl Bringmann, Marvin Künnemann, and Karol Wegrzycki. Approximating apsp without scaling: equivalence of approximate min-plus and exact min-max. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, pages 943-954, New York, NY, USA, 2019. Association for Computing Machinery. URL: https://doi.org/10.1145/3313276.3316373.
  9. Edith Cohen and Uri Zwick. All-pairs small-stretch paths. Journal of Algorithms, 38(2):335-353, 1997. URL: https://doi.org/http://doi.org/10.1006/jagm.2000.1117.
  10. Mingyang Deng, Yael Kirkpatrick, Victor Rong, Virginia Vassilevska Williams, and Ziqian Zhong. New additive approximations for shortest paths and cycles. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), volume 229 of Leibniz International Proceedings in Informatics (LIPIcs), pages 50:1-50:10, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.50.
  11. Dorit Dor, Shay Halperin, and Uri Zwick. All-pairs almost shortest paths. SIAM Journal on Computing, 29(5):1740-1759, 2000. URL: https://doi.org/10.1137/S0097539797327908.
  12. Michal Dory, Sebastian Forster, Yael Kirkpatrick, Yasamin Nazari, Virginia Vassilevska Williams, and Tijn De Vos. Fast 2-approximate all-pairs shortest paths. arXiv, 2023. URL: https://doi.org/10.1137/1.9781611977912.169.
  13. Ran Duan, Hongxun Wu, and Renfei Zhou. Faster matrix multiplication via asymmetric hashing. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 2129-2138, Los Alamitos, CA, USA, 2023. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS57990.2023.00130.
  14. Michael Elkin, Yuval Gitlitz, and Ofer Neiman. Improved weighted additive spanners. In 35th International Symposium on Distributed Computing (DISC 2021), volume 209 of Leibniz International Proceedings in Informatics (LIPIcs), pages 21:1-21:15, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.DISC.2021.21.
  15. Michael Elkin and Ofer Neiman. Centralized, parallel, and distributed multi-source shortest paths via hopsets and rectangular matrix multiplication. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022), volume 219 of Leibniz International Proceedings in Informatics (LIPIcs), pages 27:1-27:22, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2022.27.
  16. Donald Bruce Johnson. Efficient algorithms for shortest paths in sparse networks. J. ACM, 24(1):1-13, January 1977. URL: https://doi.org/10.1145/321992.321993.
  17. Telikepalli Kavitha. Faster algorithms for all-pairs small stretch distances in weighted graphs. Algorithmica, 63:224-245, 2012. URL: https://doi.org/10.1007/s00453-011-9529-y.
  18. An La and Hung Le. New weighted additive spanners. arXiv, 2024. URL: https://doi.org/10.48550/arXiv.2408.14638.
  19. Michael Lawrence Fredman. New bounds on the complexity of the shortest path problem. SIAM Journal on Computing, 5(1):83-89, 1976. URL: https://doi.org/10.1137/0205006.
  20. Liam Roditty. New algorithms for all pairs approximate shortest paths. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 309-320, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3564246.3585197.
  21. Bernard Roy. Transitivité et connexité. Comptes rendus de l'Académie des Sciences, 249:216-218, 1959. URL: https://gallica.bnf.fr/ark:/12148/bpt6k3201c/f222.image.
  22. Barna Saha and Christopher Ye. Faster Approximate All Pairs Shortest Paths, pages 4758-4827. siam, 2023. URL: https://doi.org/10.1137/1.9781611977912.170.
  23. Avi Shoshan and Uri Zwick. All pairs shortest paths in undirected graphs with integer weights. FOCS '99, page 605, USA, 1999. IEEE Computer Society. Google Scholar
  24. Tadao Takaoka. A new upper bound on the complexity of the all pairs shortest path problem. Information Processing Letters, 43(4):195-199, 1992. URL: https://doi.org/http://doi.org/10.1016/0020-0190(92)90200-F.
  25. Mikkel Thorup and Uri Zwick. Approximate distance oracles. Journal of the ACM, 52:1-24, 2005. URL: https://doi.org/10.1145/1044731.1044732.
  26. Alfred Vaino Aho, John Edward Hopcroft, and Jeffrey David Ullmand. The Design and Analysis of Computer Algorithms. Always learning. Pearson Education, 1974. URL: https://books.google.co.il/books?id=0ipKsZbwSAoC.
  27. Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. New Bounds for Matrix Multiplication: from Alpha to Omega, pages 3792-3835. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.134.
  28. Stephen Warshall. A theorem on boolean matrices. J. ACM, 9(1):11-12, January 1962. URL: https://doi.org/10.1145/321105.321107.
  29. Richard Ryan Williams. Faster All-Pairs Shortest Paths via Circuit Complexity. SIAM Journal on Computing, 47(5):1965-1985, 2018. URL: https://doi.org/10.1137/15M1024524.
  30. Robert Willoughby Floyd. Algorithm 97: Shortest path. Commun. ACM, 5(6):345, June 1962. URL: https://doi.org/10.1145/367766.368168.
  31. Uri Zwick. All pairs shortest paths using bridging sets and rectangular matrix multiplication. J. ACM, 49(3):289-317, 2002. URL: https://doi.org/10.1145/567112.567114.
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