Document Open Access Logo

Near-Optimal Differentially Private Graph Algorithms via the Multidimensional AboveThreshold Mechanism

Authors Laxman Dhulipala , Monika Henzinger , George Z. Li , Quanquan C. Liu , A. R. Sricharan , Leqi Zhu

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2025.91.pdf
  • Filesize: 0.82 MB
  • 20 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Security and privacy
  • Theory of computation
Keywords
  • differential privacy
  • abovethreshold
  • densest subgraph

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

Many differentially private and classical non-private graph algorithms rely crucially on determining whether some property of each vertex meets a threshold. For example, for the k-core decomposition problem, the classic peeling algorithm iteratively removes a vertex if its induced degree falls below a threshold. The sparse vector technique (SVT) is generally used to transform non-private threshold queries into private ones with only a small additive loss in accuracy. However, a naive application of SVT in the graph setting leads to an amplification of the error by a factor of n due to composition, as SVT is applied to every vertex. In this paper, we resolve this problem by formulating a novel generalized sparse vector technique which we call the Multidimensional AboveThreshold (MAT) Mechanism which generalizes SVT (applied to vectors with one dimension) to vectors with multiple dimensions. When applied to vectors with n dimensions, we solve a number of important graph problems with better bounds than previous work.
Specifically, we apply our MAT mechanism to obtain a set of improved bounds for a variety of problems including k-core decomposition, densest subgraph, low out-degree ordering, and vertex coloring. We give a tight local edge differentially private (LEDP) algorithm for k-core decomposition that results in an approximation with O(ε^{-1} log n) additive error and no multiplicative error in O(n) rounds. We also give a new (2+η)-factor multiplicative, O(ε^{-1} log n) additive error algorithm in O(log² n) rounds for any constant η > 0. Both of these results are asymptotically tight against our new lower bound of Ω(log n) for any constant-factor approximation algorithm for k-core decomposition. Our new algorithms for k-core decomposition also directly lead to new algorithms for the related problems of densest subgraph and low out-degree ordering. Finally, we give novel LEDP differentially private defective coloring algorithms that use number of colors given in terms of the arboricity of the graph.

Cite As Get BibTex

Laxman Dhulipala, Monika Henzinger, George Z. Li, Quanquan C. Liu, A. R. Sricharan, and Leqi Zhu. Near-Optimal Differentially Private Graph Algorithms via the Multidimensional AboveThreshold Mechanism. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 91:1-91:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.91

Author Details

Laxman Dhulipala
  • University of Maryland, College Park, MD, USA
Monika Henzinger
  • Institute of Science and Technology Austria, Klosterneuburg, Austria
George Z. Li
  • Carnegie Mellon University, Pittsburgh, PA, USA
Quanquan C. Liu
  • Yale University, New Haven, CT, USA
A. R. Sricharan
  • Faculty of Computer Science, UniVie Doctoral School Computer Science DoCS, University of Vienna, Austria
Leqi Zhu
  • University of Manitoba, Canada

Funding

Monika Henzinger and A. R. Sricharan: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (MoDynStruct, No. 101019564) and the Austrian Science Fund (FWF) grant https://www.doi.org/10.55776/Z422 and grant https://www.doi.org/10.55776/I5982. Laxman Dhulipala and George Z. Li: supported by NSF award number CNS-2317194. Quanquan C. Liu: supported by a Google Academic Research Award and by an NSF award number CCF-2453323.

References

  1. J. Ignacio Alvarez-Hamelin, Luca Dall'Asta, Alain Barrat, and Alessandro Vespignani. Large scale networks fingerprinting and visualization using the k-core decomposition. In Proceedings of the 18th International Conference on Neural Information Processing Systems, 2005. Google Scholar
  2. Sepehr Assadi, Amit Chakrabarti, Prantar Ghosh, and Manuel Stoeckl. Coloring in graph streams via deterministic and adversarially robust algorithms. In Proceedings of the 42nd ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS '23, pages 141-153, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3584372.3588681.
  3. Bahman Bahmani, Ashish Goel, and Kamesh Munagala. Efficient primal-dual graph algorithms for MapReduce. In International Workshop on Algorithms and Models for the Web Graph (WAW), volume 8882, pages 59-78, 2014. URL: https://doi.org/10.1007/978-3-319-13123-8_6.
  4. Bahman Bahmani, Ravi Kumar, and Sergei Vassilvitskii. Densest subgraph in streaming and mapreduce. Proc. VLDB Endow., 5(5):454-465, 2012. URL: https://doi.org/10.14778/2140436.2140442.
  5. Amos Beimel, Kobbi Nissim, and Eran Omri. Distributed private data analysis: Simultaneously solving how and what. In Annual International Cryptology Conference, pages 451-468, 2008. URL: https://doi.org/10.1007/978-3-540-85174-5_25.
  6. Suman K. Bera, Amit Chakrabarti, and Prantar Ghosh. Graph coloring via degeneracy in streaming and other space-conscious models. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 11:1-11:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.ICALP.2020.11.
  7. Suman K. Bera, Lior Gishboliner, Yevgeny Levanzov, C. Seshadhri, and Asaf Shapira. Counting subgraphs in degenerate graphs. J. ACM, 69(3), June 2022. URL: https://doi.org/10.1145/3520240.
  8. Suman K. Bera, Noujan Pashanasangi, and C. Seshadhri. Linear time subgraph counting, graph degeneracy, and the chasm at size six. In Innovations in Theoretical Computer Science Conference (ITCS), volume 151, pages 38:1-38:20, 2020. URL: https://doi.org/10.4230/LIPICS.ITCS.2020.38.
  9. Edvin Berglin and Gerth Stølting Brodal. A simple greedy algorithm for dynamic graph orientation. Algorithmica, 82(2):245-259, 2020. URL: https://doi.org/10.1007/S00453-018-0528-0.
  10. Anup Bhattacharya, Arijit Bishnu, Gopinath Mishra, and Anannya Upasana. Even the easiest(?) graph coloring problem is not easy in streaming! In James R. Lee, editor, 12th Innovations in Theoretical Computer Science Conference, ITCS 2021, January 6-8, 2021, Virtual Conference, volume 185 of LIPIcs, pages 15:1-15:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ITCS.2021.15.
  11. Sayan Bhattacharya, Fabrizio Grandoni, Janardhan Kulkarni, Quanquan C. Liu, and Shay Solomon. Fully dynamic (Δ +1)-coloring in O(1) update time. ACM Trans. Algorithms, 18(2):10:1-10:25, 2022. URL: https://doi.org/10.1145/3494539.
  12. Sayan Bhattacharya, Monika Henzinger, Danupon Nanongkai, and Charalampos Tsourakakis. Space- and time-efficient algorithm for maintaining dense subgraphs on one-pass dynamic streams. In ACM Symposium on Theory of Computing (STOC), pages 173-182, 2015. URL: https://doi.org/10.1145/2746539.2746592.
  13. Chandra Chekuri, Aleksander Bjørn Christiansen, Jacob Holm, Ivor van der Hoog, Kent Quanrud, Eva Rotenberg, and Chris Schwiegelshohn. Adaptive out-orientations with applications. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3062-3088. SIAM, 2024. Google Scholar
  14. Chandra Chekuri, Kent Quanrud, and Manuel R. Torres. Densest subgraph: Supermodularity, iterative peeling, and flow. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1531-1555, 2022. URL: https://doi.org/10.1137/1.9781611977073.64.
  15. Aleksander BG Christiansen, Eva Rotenberg, Teresa Anna Steiner, and Juliette Vlieghe. Private graph colouring with limited defectiveness. arXiv preprint arXiv:2404.18692, 2024. Google Scholar
  16. Luciano da Fontoura Costa, Osvaldo N Oliveira Jr, Gonzalo Travieso, Francisco Aparecido Rodrigues, Paulino Ribeiro Villas Boas, Lucas Antiqueira, Matheus Palhares Viana, and Luis Enrique Correa Rocha. Analyzing and modeling real-world phenomena with complex networks: a survey of applications. Advances in Physics, 60(3):329-412, 2011. Google Scholar
  17. Anindya De. Lower bounds in differential privacy. In Theory of Cryptography Conference, TCC, pages 321-338. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-28914-9_18.
  18. Laxman Dhulipala, Guy E. Blelloch, and Julian Shun. Julienne: A framework for parallel graph algorithms using work-efficient bucketing. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 293-304, 2017. URL: https://doi.org/10.1145/3087556.3087580.
  19. Laxman Dhulipala, Guy E. Blelloch, and Julian Shun. Theoretically efficient parallel graph algorithms can be fast and scalable. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 2018. Google Scholar
  20. Laxman Dhulipala, Quanquan C. Liu, Sofya Raskhodnikova, Jessica Shi, Julian Shun, and Shangdi Yu. Differential privacy from locally adjustable graph algorithms: k-core decomposition, low out-degree ordering, and densest subgraphs. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 754-765. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00077.
  21. Michael Dinitz, Satyen Kale, Silvio Lattanzi, and Sergei Vassilvitskii. Almost tight bounds for differentially private densest subgraph. In Yossi Azar and Debmalya Panigrahi, editors, Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025, New Orleans, LA, USA, January 12-15, 2025, pages 2908-2950. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.94.
  22. Michael Dinitz, George Z. Li, Quanquan C. Liu, and Felix Zhou. Differentially private matchings, 2025. URL: https://doi.org/10.48550/arXiv.2501.00926.
  23. Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. In Proceedings of the Third Conference on Theory of Cryptography, pages 265-284, 2006. URL: https://doi.org/10.1007/11681878_14.
  24. Cynthia Dwork, Moni Naor, Omer Reingold, Guy N Rothblum, and Salil Vadhan. On the complexity of differentially private data release: efficient algorithms and hardness results. In ACM Symposium on Theory of Computing (STOC), pages 381-390, 2009. URL: https://doi.org/10.1145/1536414.1536467.
  25. Cynthia Dwork and Aaron Roth. The algorithmic foundations of differential privacy. Foundations and Trends® in Theoretical Computer Science, 9(3-4):211-407, 2014. URL: https://doi.org/10.1561/0400000042.
  26. Talya Eden, Quanquan C. Liu, Sofya Raskhodnikova, and Adam D. Smith. Triangle counting with local edge differential privacy. In International Colloquium on Automata, Languages, and Programming, ICALP, pages 52:1-52:21, 2023. URL: https://doi.org/10.4230/LIPICS.ICALP.2023.52.
  27. Hossein Esfandiari, MohammadTaghi Hajiaghayi, and David P Woodruff. Brief announcement: Applications of uniform sampling: Densest subgraph and beyond. In Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures, pages 397-399, 2016. URL: https://doi.org/10.1145/2935764.2935813.
  28. Hossein Esfandiari, Silvio Lattanzi, and Vahab Mirrokni. Parallel and streaming algorithms for k-core decomposition. In Proceedings of the 35th International Conference on Machine Learning, pages 1397-1406, 2018. Google Scholar
  29. Alireza Farhadi, Mohammad Taghi Hajiaghai, and Elaine Shi. Differentially private densest subgraph. In International Conference on Artificial Intelligence and Statistics (AISTATS), volume 151, pages 11581-11597, 2022. URL: https://proceedings.mlr.press/v151/farhadi22a.html.
  30. Mohsen Ghaffari and Fabian Kuhn. Deterministic distributed vertex coloring: Simpler, faster, and without network decomposition. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 1009-1020. IEEE, 2022. Google Scholar
  31. Mohsen Ghaffari, Silvio Lattanzi, and Slobodan Mitrović. Improved parallel algorithms for density-based network clustering. In Proceedings of the 36th International Conference on Machine Learning, pages 2201-2210, 2019. URL: http://proceedings.mlr.press/v97/ghaffari19a.html.
  32. Mohsen Ghaffari and Christiana Lymouri. Simple and near-optimal distributed coloring for sparse graphs. In Andréa W. Richa, editor, 31st International Symposium on Distributed Computing, DISC 2017, October 16-20, 2017, Vienna, Austria, volume 91 of LIPIcs, pages 20:1-20:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPICS.DISC.2017.20.
  33. Meng He, Ganggui Tang, and Norbert Zeh. Orienting dynamic graphs, with applications to maximal matchings and adjacency queries. In International Symposium on Algorithms and Computation, pages 128-140. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-13075-0_11.
  34. Monika Henzinger, Stefan Neumann, and Andreas Wiese. Explicit and implicit dynamic coloring of graphs with bounded arboricity. CoRR, abs/2002.10142, 2020. URL: https://arxiv.org/abs/2002.10142.
  35. Monika Henzinger and Pan Peng. Constant-time dynamic (Δ +1)-coloring. ACM Trans. Algorithms, 18(2):16:1-16:21, 2022. URL: https://doi.org/10.1145/3501403.
  36. Justin Hsu, Zhiyi Huang, Aaron Roth, Tim Roughgarden, and Zhiwei Steven Wu. Private matchings and allocations. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 21-30, 2014. URL: https://doi.org/10.1145/2591796.2591826.
  37. H. Kabir and K. Madduri. Parallel k-core decomposition on multicore platforms. In IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW), pages 1482-1491, 2017. Google Scholar
  38. Łukasz Kowalik. Approximation scheme for lowest outdegree orientation and graph density measures. In International Symposium on Algorithms and Computation, pages 557-566. Springer, 2006. Google Scholar
  39. Tommaso Lanciano, Atsushi Miyauchi, Adriano Fazzone, and Francesco Bonchi. A survey on the densest subgraph problem and its variants. arXiv preprint arXiv:2303.14467, 2023. URL: https://doi.org/10.48550/arXiv.2303.14467.
  40. Yang Li, Michael Purcell, Thierry Rakotoarivelo, David Smith, Thilina Ranbaduge, and Kee Siong Ng. Private graph data release: A survey. ACM Computing Surveys, 55(11):1-39, 2023. URL: https://doi.org/10.1145/3569085.
  41. Jingcheng Liu and Kunal Talwar. Private selection from private candidates. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 298-309, 2019. URL: https://doi.org/10.1145/3313276.3316377.
  42. Quanquan C. Liu, Jessica Shi, Shangdi Yu, Laxman Dhulipala, and Julian Shun. Parallel batch-dynamic algorithms for k-core decomposition and related graph problems. In 34th ACM Symposium on Parallelism in Algorithms and Architectures, pages 191-204, 2022. URL: https://doi.org/10.1145/3490148.3538569.
  43. Fragkiskos D Malliaros, Christos Giatsidis, Apostolos N Papadopoulos, and Michalis Vazirgiannis. The core decomposition of networks: Theory, algorithms and applications. The VLDB Journal, 29:61-92, 2020. URL: https://doi.org/10.1007/S00778-019-00587-4.
  44. David W. Matula and Leland L. Beck. Smallest-last ordering and clustering and graph coloring algorithms. J. ACM, 30(3):417-427, July 1983. URL: https://doi.org/10.1145/2402.322385.
  45. Yannic Maus. Distributed graph coloring made easy. ACM Trans. Parallel Comput., 10(4), December 2023. URL: https://doi.org/10.1145/3605896.
  46. Andrew McGregor, David Tench, Sofya Vorotnikova, and Hoa T Vu. Densest subgraph in dynamic graph streams. In International Symposium on Mathematical Foundations of Computer Science, pages 472-482. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48054-0_39.
  47. Ofer Neiman and Shay Solomon. Simple deterministic algorithms for fully dynamic maximal matching. ACM Transactions on Algorithms (TALG), 12(1):1-15, 2015. URL: https://doi.org/10.1145/2700206.
  48. Dung Nguyen and Anil Vullikanti. Differentially private densest subgraph detection. In Proceedings of the 38th International Conference on Machine Learning, pages 8140-8151, 2021. URL: http://proceedings.mlr.press/v139/nguyen21i.html.
  49. Ahmet Erdem Sariyüce and Ali Pinar. Fast hierarchy construction for dense subgraphs. Proceedings of the VLDB Endowment, 10(3):97-108, 2016. URL: https://doi.org/10.14778/3021924.3021927.
  50. Saurabh Sawlani and Junxing Wang. Near-optimal fully dynamic densest subgraph. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 181-193, 2020. URL: https://doi.org/10.1145/3357713.3384327.
  51. Kijung Shin, Tina Eliassi-Rad, and Christos Faloutsos. Corescope: Graph mining using k-core analysis—patterns, anomalies and algorithms. In 2016 IEEE 16th international conference on data mining (ICDM), pages 469-478. IEEE, 2016. URL: https://doi.org/10.1109/ICDM.2016.0058.
  52. Shay Solomon and Nicole Wein. Improved dynamic graph coloring. ACM Trans. Algorithms, 16(3), June 2020. URL: https://doi.org/10.1145/3392724.
  53. Hsin-Hao Su and Hoa T. Vu. Distributed Dense Subgraph Detection and Low Outdegree Orientation. In 34th International Symposium on Distributed Computing, pages 15:1-15:18, 2020. URL: https://doi.org/10.4230/LIPICS.DISC.2020.15.
  54. Salil P. Vadhan and Tianhao Wang. Concurrent composition of differential privacy. In Kobbi Nissim and Brent Waters, editors, Theory of Cryptography - 19th International Conference, TCC 2021, Raleigh, NC, USA, November 8-11, 2021, Proceedings, Part II, volume 13043 of Lecture Notes in Computer Science, pages 582-604. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-90453-1_20.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

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