Document Open Access Logo

Streaming Algorithms for Network Design

Authors Chandra Chekuri , Rhea Jain , Sepideh Mahabadi , Ali Vakilian

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2025.4.pdf
  • Filesize: 0.96 MB
  • 23 pages
HTML5 Logo HTML (experimental)
LIPIcs.APPROX-RANDOM.2025.4.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Routing and network design problems
Keywords
  • Streaming Algorithms
  • Survivable Network Design
  • Fault-Tolerant Spanners

Metrics

Abstract

We consider the Survivable Network Design problem (SNDP) in the single-pass insertion-only streaming model. The input to SNDP is an edge-weighted graph G = (V, E) and an integer connectivity requirement r(uv) for each u, v ∈ V. The objective is to find a minimum-weight subgraph H ⊆ G such that, for every pair of vertices u, v ∈ V, u and v are r(uv)-edge/vertex-connected. Recent work by [Ce Jin et al., 2024] obtained approximation algorithms for edge-connectivity augmentation, and via that, also derived algorithms for edge-connectivity SNDP (EC-SNDP). In this work we consider vertex-connectivity setting (VC-SNDP) and obtain several results for it as well as improved results for EC-SNDP.  
- We provide a general framework for solving connectivity problems including SNDP and others in streaming; this is based on a connection to fault-tolerant spanners. For VC-SNDP we provide an O(tk)-approximation in Õ(k^{1-1/t}n^{1 + 1/t}) space, where k is the maximum connectivity requirement, assuming an exact algorithm at the end of the stream. Using a refined LP-based analysis, we provide an O(β t)-approximation where β is the integrality gap of the natural cut-based LP relaxation. These are the first approximation algorithms in the streaming model for VC-SNDP. When applied to the EC-SNDP, our framework provides an O(t)-approximation in Õ(k^{1/2-1/(2t)}n^{1 + 1/t} + kn) space, improving the O(t log k)-approximation of [Ce Jin et al., 2024] using Õ(kn^{1+1/t}) space; this also extends to element-connectivity SNDP. 
- We consider vertex connectivity-augmentation in the link-arrival model. The input is a k-vertex-connected spanning subgraph G, and additional weighted links L arrive in the stream; the goal is to store the min-weight set of links such that G ∪ L is (k+1)-vertex-connected. We obtain constant-factor approximations in near-linear space for k = 1, 2. Our result for k = 2 is based on using the SPQR tree, a novel application for this well-known representation of 2-connected graphs.

Cite As Get BibTex

Chandra Chekuri, Rhea Jain, Sepideh Mahabadi, and Ali Vakilian. Streaming Algorithms for Network Design. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 4:1-4:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.4

Author Details

Chandra Chekuri
  • Siebel School of Computing and Data Science, University of Illinois, Urbana, IL, USA
Rhea Jain
  • Siebel School of Computing and Data Science, University of Illinois, Urbana, IL, USA
Sepideh Mahabadi
  • Microsoft Research, Redmond, WA, USA
Ali Vakilian
  • Toyota Technological Institute at Chicago, IL, USA

Funding

  • Chekuri, Chandra: Supported in part by NSF grant CCF-2402667.
  • Jain, Rhea: Supported in part by NSF grant CCF-2402667.

Acknowledgements

This work was conducted in part while Chandra Chekuri, Sepideh Mahabadi and Ali Vakilian were visitors at the Simons Institute for the Theory of Computing as part of the Data Structures and Optimization for Fast Algorithms program. The authors thank Ce Jin for his contributions during the early stage of the project. The authors also thank an anonymous reviewer for pointers to efficient constructions and improved bounds for fault-tolerant spanners, and for pointing out an error in the previous version.

References

  1. Kook Jin Ahn and Sudipto Guha. Graph sparsification in the semi-streaming model. In International Colloquium on Automata, Languages, and Programming, pages 328-338. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-02930-1_27.
  2. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Graph sketches: sparsification, spanners, and subgraphs. In Symposium on Principles of Database Systems (PODS), pages 5-14, 2012. URL: https://doi.org/10.1145/2213556.2213560.
  3. Ingo Althöfer, Gautam Das, David Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9(1):81-100, 1993. URL: https://doi.org/10.1007/BF02189308.
  4. Sepehr Assadi and Aditi Dudeja. A simple semi-streaming algorithm for global minimum cuts. In Symposium on Simplicity in Algorithms (SOSA), pages 172-180. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976496.19.
  5. Sepehr Assadi, Sanjeev Khanna, and Yang Li. On estimating maximum matching size in graph streams. In Proceedings of the Symposium on Discrete Algorithms (SODA), pages 1723-1742, 2017. URL: https://doi.org/10.1137/1.9781611974782.113.
  6. Sepehr Assadi, Sanjeev Khanna, Yang Li, and Grigory Yaroslavtsev. Maximum matchings in dynamic graph streams and the simultaneous communication model. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms (SODA), pages 1345-1364, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH93.
  7. Sepehr Assadi and Vihan Shah. Tight bounds for vertex connectivity in dynamic streams. In Symposium on Simplicity in Algorithms (SOSA), pages 213-227. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977585.CH20.
  8. Surender Baswana. Streaming algorithm for graph spanners-single pass and constant processing time per edge. Inf. Process. Lett., 106(3):110-114, 2008. URL: https://doi.org/10.1016/J.IPL.2007.11.001.
  9. Greg Bodwin, Michael Dinitz, Merav Parter, and Virginia Vassilevska Williams. Optimal vertex fault tolerant spanners (for fixed stretch). In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1884-1900. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.123.
  10. Greg Bodwin, Michael Dinitz, and Caleb Robelle. Optimal vertex fault-tolerant spanners in polynomial time. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2924-2938. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.174.
  11. Greg Bodwin, Michael Dinitz, and Caleb Robelle. Partially optimal edge fault-tolerant spanners. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3272-3286. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.129.
  12. Greg Bodwin and Shyamal Patel. A trivial yet optimal solution to vertex fault tolerant spanners. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing (PODS), pages 541-543, 2019. URL: https://doi.org/10.1145/3293611.3331588.
  13. Jarosław Byrka, Fabrizio Grandoni, and Afrouz Jabal Ameli. Breaching the 2-approximation barrier for connectivity augmentation: a reduction to steiner tree. In Symposium on Theory of Computing (STOC), pages 815-825, 2020. URL: https://doi.org/10.1145/3357713.3384301.
  14. Jarosław Byrka, Fabrizio Grandoni, Thomas Rothvoß, and Laura Sanità. Steiner tree approximation via iterative randomized rounding. Journal of the ACM (JACM), 60(1):1-33, 2013. URL: https://doi.org/10.1145/2432622.2432628.
  15. Federica Cecchetto, Vera Traub, and Rico Zenklusen. Bridging the gap between tree and connectivity augmentation: unified and stronger approaches. In Symposium on Theory of Computing (STOC), pages 370-383, 2021. URL: https://doi.org/10.1145/3406325.3451086.
  16. Tanmoy Chakraborty, Julia Chuzhoy, and Sanjeev Khanna. Network design for vertex connectivity. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 167-176, 2008. URL: https://doi.org/10.1145/1374376.1374403.
  17. Chandra Chekuri, Rhea Jain, Sepideh Mahabadi, and Ali Vakilian. Streaming algorithms for network design, 2025. URL: https://doi.org/10.48550/arXiv.2503.00712.
  18. Joseph Cheriyan, Tibor Jordán, and Ramamoorthi Ravi. On 2-coverings and 2-packings of laminar families. In 7th Annual European Symposium (ESA), pages 510-520. Springer, 1999. Google Scholar
  19. Joseph Cheriyan, Ming-Yang Kao, and Ramakrishna Thurimella. Scan-first search and sparse certificates: an improved parallel algorithm for k-vertex connectivity. SIAM Journal on Computing, 22(1):157-174, 1993. URL: https://doi.org/10.1137/0222013.
  20. Joseph Cheriyan and László A Végh. Approximating minimum-cost k-node connected subgraphs via independence-free graphs. SIAM Journal on Computing, 43(4):1342-1362, 2014. URL: https://doi.org/10.1137/120902847.
  21. Joseph Cheriyan, Santosh Vempala, and Adrian Vetta. Network design via iterative rounding of setpair relaxations. Combinatorica, 26(3):255-275, 2006. URL: https://doi.org/10.1007/S00493-006-0016-Z.
  22. Julia Chuzhoy and Sanjeev Khanna. An O(k³ log n)-approximation algorithm for vertex-connectivity survivable network design. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 437-441, 2009. URL: https://doi.org/10.1109/FOCS.2009.38.
  23. Michael S. Crouch, Andrew McGregor, and Daniel M. Stubbs. Dynamic graphs in the sliding-window model. In Algorithms - ESA 2013 - 21st Annual European Symposium, volume 8125 of Lecture Notes in Computer Science, pages 337-348. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-40450-4_29.
  24. Artur Czumaj, Shaofeng H-C Jiang, Robert Krauthgamer, and Pavel Veselỳ. Streaming algorithms for geometric steiner forest. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), 2022. Google Scholar
  25. Giuseppe Di Battista and Roberto Tamassia. Incremental planarity testing. In 30th Annual Symposium on Foundations of Computer Science (FOCS), pages 436-441, 1989. Google Scholar
  26. Giuseppe Di Battista and Roberto Tamassia. On-line maintenance of triconnected components with spqr-trees. Algorithmica, 15(4):302-318, 1996. URL: https://doi.org/10.1007/BF01961541.
  27. Giuseppe Di Battista and Roberto Tamassia. On-line planarity testing. SIAM Journal on Computing, 25(5):956-997, 1996. URL: https://doi.org/10.1137/S0097539794280736.
  28. E A Dinits, Alexander V Karzanov, and Micael V Lomonosov. On the structure of a family of minimal weighted cuts in a graph. Studies in Discrete Optimization, pages 290-306, 1973. Google Scholar
  29. Michael Dinitz and Caleb Robelle. Efficient and simple algorithms for fault-tolerant spanners. In Proceedings of the 39th Symposium on Principles of Distributed Computing, pages 493-500, 2020. URL: https://doi.org/10.1145/3382734.3405735.
  30. Michael Elkin. Streaming and fully dynamic centralized algorithms for constructing and maintaining sparse spanners. ACM Trans. Algorithms, 7(2):20:1-20:17, 2011. URL: https://doi.org/10.1145/1921659.1921666.
  31. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. Theoretical Computer Science, 348(2-3):207-216, 2005. URL: https://doi.org/10.1016/J.TCS.2005.09.013.
  32. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. Graph distances in the data-stream model. Journal on Computing, 38(5):1709-1727, 2008. URL: https://doi.org/10.1137/070683155.
  33. Manuel Fernández V, David P Woodruff, and Taisuke Yasuda. Graph spanners in the message-passing model. In Innovations in Theoretical Computer Science Conference, 2020. Google Scholar
  34. Arnold Filtser, Michael Kapralov, and Navid Nouri. Graph spanners by sketching in dynamic streams and the simultaneous communication model. In Proceedings of the Symposium on Discrete Algorithms (SODA), pages 1894-1913, 2021. URL: https://doi.org/10.1137/1.9781611976465.113.
  35. Lisa Fleischer, Kamal Jain, and David P Williamson. Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems. Journal of Computer and System Sciences, 72(5):838-867, 2006. URL: https://doi.org/10.1016/J.JCSS.2005.05.006.
  36. Takuro Fukunaga, Zeev Nutov, and R Ravi. Iterative rounding approximation algorithms for degree-bounded node-connectivity network design. SIAM Journal on Computing, 44(5):1202-1229, 2015. URL: https://doi.org/10.1137/13094503X.
  37. Mohit Garg, Fabrizio Grandoni, and Afrouz Jabal Ameli. Improved approximation for two-edge-connectivity. In Symposium on Discrete Algorithms (SODA), pages 2368-2410, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH92.
  38. Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching. In Symposium on Discrete Algorithms (SODA), pages 468-485, 2012. URL: https://doi.org/10.1137/1.9781611973099.41.
  39. Michel X. Goemans, Andrew V. Goldberg, Serge A. Plotkin, David B. Shmoys, Éva Tardos, and David P. Williamson. Improved approximation algorithms for network design problems. In Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 223-232. ACM/SIAM, 1994. URL: http://dl.acm.org/citation.cfm?id=314464.314497.
  40. Sudipto Guha, Andrew McGregor, and David Tench. Vertex and hyperedge connectivity in dynamic graph streams. In Proceedings of the 34th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pages 241-247, 2015. URL: https://doi.org/10.1145/2745754.2745763.
  41. Venkatesan Guruswami and Krzysztof Onak. Superlinear lower bounds for multipass graph processing. Algorithmica, 76:654-683, 2016. URL: https://doi.org/10.1007/S00453-016-0138-7.
  42. Carsten Gutwenger and Petra Mutzel. A linear time implementation of spqr-trees. In International Symposium on Graph Drawing, pages 77-90. Springer, 2000. URL: https://doi.org/10.1007/3-540-44541-2_8.
  43. J. E. Hopcroft and R. E. Tarjan. Dividing a graph into triconnected components. SIAM Journal on Computing, 2(3):135-158, 1973. URL: https://doi.org/10.1137/0202012.
  44. Kamal Jain. A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica, 21:39-60, 2001. URL: https://doi.org/10.1007/S004930170004.
  45. Ce Jin, Michael Kapralov, Sepideh Mahabadi, and Ali Vakilian. Streaming algorithms for connectivity augmentation. In 51st International Colloquium on Automata, Languages, and Programming (ICALP), volume 297 of LIPIcs, pages 93:1-93:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.ICALP.2024.93.
  46. Michael Kapralov. Space lower bounds for approximating maximum matching in the edge arrival model. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1874-1893, 2021. URL: https://doi.org/10.1137/1.9781611976465.112.
  47. Michael Kapralov, Sanjeev Khanna, and Madhu Sudan. Streaming lower bounds for approximating max-cut. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1263-1282. SIAM, 2014. Google Scholar
  48. Michael Kapralov, Yin Tat Lee, Cameron Musco, Christopher Musco, and Aaron Sidford. Single pass spectral sparsification in dynamic streams. SIAM J. Comput., 46(1):456-477, 2017. URL: https://doi.org/10.1137/141002281.
  49. Michael Kapralov, Aida Mousavifar, Cameron Musco, Christopher Musco, Navid Nouri, Aaron Sidford, and Jakab Tardos. Fast and space efficient spectral sparsification in dynamic streams. In Proceedings of the Symposium on Discrete Algorithms (SODA), pages 1814-1833, 2020. URL: https://doi.org/10.1137/1.9781611975994.111.
  50. Michael Kapralov and David Woodruff. Spanners and sparsifiers in dynamic streams. In Proceedings of the Symposium on Principles of Distributed Computing, pages 272-281, 2014. URL: https://doi.org/10.1145/2611462.2611497.
  51. Jonathan A Kelner and Alex Levin. Spectral sparsification in the semi-streaming setting. Theory of Computing Systems, 53(2):243-262, 2013. URL: https://doi.org/10.1007/S00224-012-9396-1.
  52. Samir Khuller and Ramakrishna Thurimella. Approximation algorithms for graph augmentation. Journal of Algorithms, 14(2):214-225, 1993. URL: https://doi.org/10.1006/JAGM.1993.1010.
  53. Samir Khuller and Uzi Vishkin. Biconnectivity approximations and graph carvings. J. ACM, 41(2):214-235, 1994. URL: https://doi.org/10.1145/174652.174654.
  54. Bundit Laekhanukit. An improved approximation algorithm for the minimum cost subset k-connected subgraph problem. Algorithmica, 72(3):714-733, 2015. URL: https://doi.org/10.1007/S00453-014-9869-5.
  55. Christos Levcopoulos, Giri Narasimhan, and Michiel Smid. Efficient algorithms for constructing fault-tolerant geometric spanners. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 186-195, 1998. URL: https://doi.org/10.1145/276698.276734.
  56. Andrew McGregor. Finding graph matchings in data streams. In International Workshop on Approximation Algorithms for Combinatorial Optimization, pages 170-181. Springer, 2005. URL: https://doi.org/10.1007/11538462_15.
  57. Andrew McGregor. Graph stream algorithms: a survey. ACM SIGMOD Record, 43(1):9-20, 2014. URL: https://doi.org/10.1145/2627692.2627694.
  58. Peter Bro Miltersen, Noam Nisan, Shmuel Safra, and Avi Wigderson. On data structures and asymmetric communication complexity. J. Comput. Syst. Sci., 57(1):37-49, 1998. URL: https://doi.org/10.1006/JCSS.1998.1577.
  59. Hiroshi Nagamochi and Toshihide Ibaraki. Linear time algorithms for finding k-edge connected and k-node connected spanning subgraphs. Algorithmica, 7(1992):583-596, 1992. Google Scholar
  60. Jelani Nelson and Huacheng Yu. Optimal lower bounds for distributed and streaming spanning forest computation. In Proceedings of the Symposium on Discrete Algorithms (SODA), pages 1844-1860, 2019. URL: https://doi.org/10.1137/1.9781611975482.111.
  61. Z. Nutov. Approximating minimum-cost connectivity problems via uncrossable bifamilies. ACM Transactions on Algorithms (TALG), 9(1):1, 2012. Google Scholar
  62. Zeev Nutov. Approximating minimum-cost edge-covers of crossing biset-families. Combinatorica, 34(1):95-114, 2014. URL: https://doi.org/10.1007/S00493-014-2773-4.
  63. Zeev Nutov. Erratum: Approximating minimum-cost connectivity problems via uncrossable bifamilies. ACM Transactions on Algorithms (TALG), 14(3):1-8, 2018. URL: https://doi.org/10.1145/3186991.
  64. Zeev Nutov. Improved approximation algorithms for minimum cost node-connectivity augmentation problems. Theory of Computing Systems, 62:510-532, 2018. URL: https://doi.org/10.1007/S00224-017-9786-5.
  65. Zeev Nutov. A 4+ε approximation for k-connected subgraphs. Journal of Computer and System Sciences, 123:64-75, 2022. Google Scholar
  66. Xiaoming Sun and David P Woodruff. Tight bounds for graph problems in insertion streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. Google Scholar
  67. Vera Traub and Rico Zenklusen. A better-than-2 approximation for weighted tree augmentation. In Foundations of Computer Science (FOCS), pages 1-12, 2022. Google Scholar
  68. Vera Traub and Rico Zenklusen. Local search for weighted tree augmentation and steiner tree. In Symposium on Discrete Algorithms (SODA), pages 3253-3272, 2022. URL: https://doi.org/10.1137/1.9781611977073.128.
  69. Vera Traub and Rico Zenklusen. A (1.5+ ε)-approximation algorithm for weighted connectivity augmentation. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 1820-1833, 2023. URL: https://doi.org/10.1145/3564246.3585122.
  70. David P. Williamson, Michel X. Goemans, Milena Mihail, and Vijay V. Vazirani. A primal-dual approximation algorithm for generalized steiner network problems. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, STOC '93, pages 708-717, New York, NY, USA, 1993. Association for Computing Machinery. URL: https://doi.org/10.1145/167088.167268.
  71. Mariano Zelke. k-connectivity in the semi-streaming model. arXiv preprint cs/0608066, 2006. Google Scholar
  72. Mariano Zelke. Intractability of min-and max-cut in streaming graphs. Information Processing Letters, 111(3):145-150, 2011. URL: https://doi.org/10.1016/J.IPL.2010.10.017.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact