Document Open Access Logo

Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams

Authors Jacques Dark , Christian Konrad

PDF
Thumbnail PDF

File

LIPIcs.CCC.2020.30.pdf
  • Filesize: 0.56 MB
  • 14 pages

Document Identifiers

Author Details

Jacques Dark
  • Department of Computer Science, University of Warwick, Coventry, UK
Christian Konrad
  • Department of Computer Science, University of Bristol, UK

Funding

  • Dark, Jacques: Jacques Dark was supported by an EMEA Microsoft Research studentship and European Research Council grant ERC-2014-CoG 647557.

Acknowledgements

The authors are grateful for the numerous excellent comments and suggestions from an anonymous reviewer.

Cite As Get BibTex

Jacques Dark and Christian Konrad. Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.CCC.2020.30

Abstract

In this paper, we give simple optimal lower bounds on the one-way two-party communication complexity of approximate Maximum Matching and Minimum Vertex Cover with deletions. In our model, Alice holds a set of edges and sends a single message to Bob. Bob holds a set of edge deletions, which form a subset of Alice’s edges, and needs to report a large matching or a small vertex cover in the graph spanned by the edges that are not deleted. Our results imply optimal space lower bounds for insertion-deletion streaming algorithms for Maximum Matching and Minimum Vertex Cover. 
Previously, Assadi et al. [SODA 2016] gave an optimal space lower bound for insertion-deletion streaming algorithms for Maximum Matching via the simultaneous model of communication. Our lower bound is simpler and stronger in several aspects: The lower bound of Assadi et al. only holds for algorithms that (1) are able to process streams that contain a triple exponential number of deletions in n, the number of vertices of the input graph; (2) are able to process multi-graphs; and (3) never output edges that do not exist in the input graph when the randomized algorithm errs. In contrast, our lower bound even holds for algorithms that (1) rely on short (O(n²)-length) input streams; (2) are only able to process simple graphs; and (3) may output non-existing edges when the algorithm errs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming models
  • Theory of computation → Communication complexity
  • Theory of computation → Lower bounds and information complexity
Keywords
  • Maximum matching
  • Minimum vertex cover
  • Dynamic graph streams
  • Communication complexity
  • Lower bounds

Metrics

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

References

  1. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing graph structure via linear measurements. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’12, page 459–467, USA, 2012. Society for Industrial and Applied Mathematics. Google Scholar
  2. Yuqing Ai, Wei Hu, Yi Li, and David P. Woodruff. New characterizations in turnstile streams with applications. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 20:1-20:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.20.
  3. Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Sublinear algorithms for (Δ + 1) vertex coloring. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 767-786. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.48.
  4. Sepehr Assadi and Sanjeev Khanna. Randomized composable coresets for matching and vertex cover. In Proceedings of the 29th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA ’17, page 3–12, New York, NY, USA, 2017. Association for Computing Machinery. URL: https://doi.org/10.1145/3087556.3087581.
  5. Sepehr Assadi, Sanjeev Khanna, Yang Li, and Grigory Yaroslavtsev. Maximum matchings in dynamic graph streams and the simultaneous communication model. 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 1345-1364. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch93.
  6. Ziv Bar-Yossef, Thathachar S Jayram, Robert Krauthgamer, and Ravi Kumar. The sketching complexity of pattern matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 261-272. Springer, 2004. Google Scholar
  7. Suman K. Bera and Amit Chakrabarti. Towards tighter space bounds for counting triangles and other substructures in graph streams. In Heribert Vollmer and Brigitte Vallée, editors, 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017, March 8-11, 2017, Hannover, Germany, volume 66 of LIPIcs, pages 11:1-11:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.STACS.2017.11.
  8. Rajesh Chitnis, Graham Cormode, Hossein Esfandiari, MohammadTaghi Hajiaghayi, Andrew McGregor, Morteza Monemizadeh, and Sofya Vorotnikova. Kernelization via sampling with applications to finding matchings and related problems in dynamic graph streams. 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 1326-1344. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch92.
  9. Graham Cormode, Jacques Dark, and Christian Konrad. Approximating the caro-wei bound for independent sets in graph streams. In Jon Lee, Giovanni Rinaldi, and Ali Ridha Mahjoub, editors, Combinatorial Optimization - 5th International Symposium, ISCO 2018, Marrakesh, Morocco, April 11-13, 2018, Revised Selected Papers, volume 10856 of Lecture Notes in Computer Science, pages 101-114. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-96151-4_9.
  10. Graham Cormode, Jacques Dark, and Christian Konrad. Independent sets in vertex-arrival streams. 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 45:1-45:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.45.
  11. Graham Cormode and Hossein Jowhari. A second look at counting triangles in graph streams (corrected). Theor. Comput. Sci., 683:22-30, 2017. URL: https://doi.org/10.1016/j.tcs.2016.06.020.
  12. Alireza Farhadi, MohammadTaghi Hajiaghayi, Tung Mai, Anup Rao, and Ryan A. Rossi. Approximate maximum matching in random streams. In Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’20, page 1773–1785, USA, 2020. Society for Industrial and Applied Mathematics. Google Scholar
  13. Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching. In Yuval Rabani, editor, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 468-485. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.41.
  14. Bjarni V. Halldórsson, Magnús M. Halldórsson, Elena Losievskaja, and Mario Szegedy. Streaming algorithms for independent sets in sparse hypergraphs. Algorithmica, 76(2):490-501, 2016. URL: https://doi.org/10.1007/s00453-015-0051-5.
  15. Magnús M. Halldórsson, Xiaoming Sun, Mario Szegedy, and Chengu Wang. Streaming and communication complexity of clique approximation. In Artur Czumaj, Kurt Mehlhorn, Andrew M. Pitts, and Roger Wattenhofer, editors, Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part I, volume 7391 of Lecture Notes in Computer Science, pages 449-460. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-31594-7_38.
  16. Monika Rauch Henzinger, Prabhakar Raghavan, and Sridhar Rajagopalan. Computing on data streams. In James M. Abello and Jeffrey Scott Vitter, editors, External Memory Algorithms, Proceedings of a DIMACS Workshop, New Brunswick, New Jersey, USA, May 20-22, 1998, volume 50 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 107-118. DIMACS/AMS, 1998. URL: https://doi.org/10.1090/dimacs/050/05.
  17. Kaave Hosseini, Shachar Lovett, and Grigory Yaroslavtsev. Optimality of Linear Sketching Under Modular Updates. In Amir Shpilka, editor, 34th Computational Complexity Conference (CCC 2019), volume 137 of Leibniz International Proceedings in Informatics (LIPIcs), pages 13:1-13:17, Dagstuhl, Germany, 2019. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.13.
  18. Sagar Kale and Sumedh Tirodkar. Maximum matching in two, three, and a few more passes over graph streams. In Klaus Jansen, José D. P. Rolim, David Williamson, and Santosh S. Vempala, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2017, August 16-18, 2017, Berkeley, CA, USA, volume 81 of LIPIcs, pages 15:1-15:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.15.
  19. John Kallaugher and Eric Price. Separations and equivalences between turnstile streaming and linear sketching. In Symposium on Theory of Computing, STOC 2020, 2020. to appear. Google Scholar
  20. Daniel M. Kane, Kurt Mehlhorn, Thomas Sauerwald, and He Sun. Counting arbitrary subgraphs in data streams. In Artur Czumaj, Kurt Mehlhorn, Andrew M. Pitts, and Roger Wattenhofer, editors, Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part II, volume 7392 of Lecture Notes in Computer Science, pages 598-609. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-31585-5_53.
  21. Michael Kapralov, Sanjeev Khanna, and Madhu Sudan. Approximating matching size from random streams. 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 734-751. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.55.
  22. 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 Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1814-1833. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.111.
  23. Christian Konrad. Maximum matching in turnstile streams. In Nikhil Bansal and Irene Finocchi, editors, Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, volume 9294 of Lecture Notes in Computer Science, pages 840-852. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48350-3_70.
  24. Christian Konrad. A simple augmentation method for matchings with applications to streaming algorithms. In Igor Potapov, Paul G. Spirakis, and James Worrell, editors, 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018, August 27-31, 2018, Liverpool, UK, volume 117 of LIPIcs, pages 74:1-74:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.MFCS.2018.74.
  25. Christian Konrad. Streaming frequent items with timestamps and detecting large neighborhoods in graph streams. CoRR, abs/1911.08832, 2019. URL: http://arxiv.org/abs/1911.08832.
  26. Christian Konrad, Frédéric Magniez, and Claire Mathieu. Maximum matching in semi-streaming with few passes. In Anupam Gupta, Klaus Jansen, José D. P. Rolim, and Rocco A. Servedio, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Cambridge, MA, USA, August 15-17, 2012. Proceedings, volume 7408 of Lecture Notes in Computer Science, pages 231-242. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-32512-0_20.
  27. Christian Konrad and Adi Rosén. Approximating semi-matchings in streaming and in two-party communication. ACM Trans. Algorithms, 12(3):32:1-32:21, 2016. URL: https://doi.org/10.1145/2898960.
  28. Yi Li, Huy L. Nguyen, and David P. Woodruff. Turnstile streaming algorithms might as well be linear sketches. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 174-183. ACM, 2014. URL: https://doi.org/10.1145/2591796.2591812.
  29. Andrew McGregor. Graph stream algorithms: a survey. SIGMOD Record, 43(1):9-20, 2014. URL: https://doi.org/10.1145/2627692.2627694.
  30. Peter Bro Miltersen, Noam Nisan, Shmuel Safra, and Avi Wigderson. On data structures and asymmetric communication complexity. Journal of Computer and System Sciences, 57(1):37-49, 1998. Google Scholar
  31. Ami Paz and Gregory Schwartzman. A (2+ε)-approximation for maximum weight matching in the semi-streaming model. ACM Trans. Algorithms, 15(2):18:1-18:15, 2019. URL: https://doi.org/10.1145/3274668.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact