Document Open Access Logo

Space Optimal Vertex Cover in Dynamic Streams

Authors Kheeran K. Naidu , Vihan Shah



PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2022.53.pdf
  • Filesize: 0.81 MB
  • 15 pages

Document Identifiers

Author Details

Kheeran K. Naidu
  • Department of Computer Science, University of Bristol, UK
Vihan Shah
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA

Acknowledgements

We are grateful to Sepehr Assadi and Christian Konrad for many helpful discussions. We also appreciate the valuable comments from our APPROX 2022 reviewers.

Cite AsGet BibTex

Kheeran K. Naidu and Vihan Shah. Space Optimal Vertex Cover in Dynamic Streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 53:1-53:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.53

Abstract

We optimally resolve the space complexity for the problem of finding an α-approximate minimum vertex cover (αMVC) in dynamic graph streams. We give a randomised algorithm for αMVC which uses O(n²/α²) bits of space matching Dark and Konrad’s lower bound [CCC 2020] up to constant factors. By computing a random greedy matching, we identify "easy" instances of the problem which can trivially be solved by returning the entire vertex set. The remaining "hard" instances, then have sparse induced subgraphs which we exploit to get our space savings and solve αMVC. Achieving this type of optimality result is crucial for providing a complete understanding of a problem, and it has been gaining interest within the dynamic graph streaming community. For connectivity, Nelson and Yu [SODA 2019] improved the lower bound showing that Ω(n log³ n) bits of space is necessary while Ahn, Guha, and McGregor [SODA 2012] have shown that O(n log³ n) bits is sufficient. For finding an α-approximate maximum matching, the upper bound was improved by Assadi and Shah [ITCS 2022] showing that O(n²/α³) bits is sufficient while Dark and Konrad [CCC 2020] have shown that Ω(n²/α³) bits is necessary. The space complexity, however, remains unresolved for many other dynamic graph streaming problems where further improvements can still be made.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Graph algorithms analysis
Keywords
  • Graph Streaming Algorithms
  • Vertex Cover
  • Dynamic Streams
  • Approximation Algorithm

Metrics

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

References

  1. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing graph structure via linear measurements. 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 459-467. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.40.
  2. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Graph sketches: sparsification, spanners, and subgraphs. In Michael Benedikt, Markus Krötzsch, and Maurizio Lenzerini, editors, Proceedings of the 31st ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2012, Scottsdale, AZ, USA, May 20-24, 2012, pages 5-14. ACM, 2012. URL: https://doi.org/10.1145/2213556.2213560.
  3. KookJin Ahn, Graham Cormode, Sudipto Guha, Andrew McGregor, and Anthony Wirth. Correlation clustering in data streams. In International Conference on Machine Learning, pages 2237-2246. PMLR, 2015. Google Scholar
  4. Sepehr Assadi. A two-pass (conditional) lower bound for semi-streaming maximum matching. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9-12, 2022, pages 708-742. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.32.
  5. Sepehr Assadi, MohammadHossein Bateni, Aaron Bernstein, Vahab S. Mirrokni, and Cliff Stein. Coresets meet EDCS: algorithms for matching and vertex cover on massive graphs. 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 1616-1635. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.98.
  6. Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Polynomial pass lower bounds for graph streaming algorithms. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 265-276. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316361.
  7. Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Sublinear algorithms for (δ+ 1) vertex coloring. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 767-786. SIAM, 2019. Google Scholar
  8. Sepehr Assadi, Arun Jambulapati, Yujia Jin, Aaron Sidford, and Kevin Tian. Semi-streaming bipartite matching in fewer passes and optimal space. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9-12, 2022, pages 627-669. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.29.
  9. 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.
  10. Sepehr Assadi, Gillat Kol, Raghuvansh R. Saxena, and Huacheng Yu. Multi-pass graph streaming lower bounds for cycle counting, max-cut, matching size, and other problems. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 354-364. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00041.
  11. Sepehr Assadi and Vihan Shah. An asymptotically optimal algorithm for maximum matching in dynamic streams. In Mark Braverman, editor, 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 31 - February 3, 2022, Berkeley, CA, USA, volume 215 of LIPIcs, pages 9:1-9:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.9.
  12. Aaron Bernstein. Improved bounds for matching in random-order streams. 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 12:1-12:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.12.
  13. Leyla Biabani, Mark de Berg, and Morteza Monemizadeh. Maximum-weight matching in sliding windows and beyond. In Hee-Kap Ahn and Kunihiko Sadakane, editors, 32nd International Symposium on Algorithms and Computation, ISAAC 2021, December 6-8, 2021, Fukuoka, Japan, volume 212 of LIPIcs, pages 73:1-73:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.73.
  14. Vladimir Braverman, Jonathan Katzman, Charles Seidell, and Gregory Vorsanger. An optimal algorithm for large frequency moments using o(n^(1-2/k)) bits. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2014. Google Scholar
  15. 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.
  16. Michael S. Crouch, Andrew McGregor, and Daniel M. Stubbs. Dynamic graphs in the sliding-window model. In Hans L. Bodlaender and Giuseppe F. Italiano, editors, Algorithms - ESA 2013 - 21st Annual European Symposium, Sophia Antipolis, France, September 2-4, 2013. Proceedings, 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.
  17. Michael S. Crouch and Daniel M. Stubbs. Improved streaming algorithms for weighted matching, via unweighted matching. In Klaus Jansen, José D. P. Rolim, Nikhil R. Devanur, and Cristopher Moore, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2014, September 4-6, 2014, Barcelona, Spain, volume 28 of LIPIcs, pages 96-104. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2014. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.96.
  18. Jacques Dark and Christian Konrad. Optimal lower bounds for matching and vertex cover in dynamic graph streams. In Shubhangi Saraf, editor, 35th Computational Complexity Conference, CCC 2020, July 28-31, 2020, Saarbrücken, Germany (Virtual Conference), volume 169 of LIPIcs, pages 30:1-30:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.30.
  19. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. In Josep Diaz, Juhani Karhumäki, Arto Lepistö, and Donald Sannella, editors, Automata, Languages and Programming: 31st International Colloquium, ICALP 2004, Turku, Finland, July 12-16, 2004. Proceedings, volume 3142 of Lecture Notes in Computer Science, pages 531-543. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-27836-8_46.
  20. Buddhima Gamlath, Sagar Kale, Slobodan Mitrovic, and Ola Svensson. Weighted matchings via unweighted augmentations. In Peter Robinson and Faith Ellen, editors, Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, PODC 2019, Toronto, ON, Canada, July 29 - August 2, 2019, pages 491-500. ACM, 2019. URL: https://doi.org/10.1145/3293611.3331603.
  21. Piotr Indyk, Eric Price, and David P Woodruff. On the power of adaptivity in sparse recovery. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 285-294. IEEE, 2011. Google Scholar
  22. Hossein Jowhari, Mert Saglam, and Gábor Tardos. Tight bounds for lp samplers, finding duplicates in streams, and related problems. In Maurizio Lenzerini and Thomas Schwentick, editors, Proceedings of the 30th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2011, June 12-16, 2011, Athens, Greece, pages 49-58. ACM, 2011. URL: https://doi.org/10.1145/1989284.1989289.
  23. Michael Kapralov, Yin Tat Lee, Cameron Musco, Christopher Musco, and Aaron Sidford. Single pass spectral sparsification in dynamic streams. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 561-570. IEEE Computer Society, 2014. URL: https://doi.org/10.1109/FOCS.2014.66.
  24. 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.
  25. Michael Kapralov, Jelani Nelson, Jakub Pachocki, Zhengyu Wang, David P. Woodruff, and Mobin Yahyazadeh. Optimal lower bounds for universal relation, and for samplers and finding duplicates in streams. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 475-486. IEEE Computer Society, 2017. Google Scholar
  26. Sanjeev Khanna and Christian Konrad. Optimal bounds for dominating set in graph streams. In Mark Braverman, editor, 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 31 - February 3, 2022, Berkeley, CA, USA, volume 215 of LIPIcs, pages 93:1-93:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.93.
  27. 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.
  28. 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.
  29. Christian Konrad. Mis in the congested clique model in o(log log δ) rounds. arXiv preprint, 2018. URL: http://arxiv.org/abs/1802.07647.
  30. Christian Konrad. Frequent elements with witnesses in data streams. In Proceedings of the 40th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pages 83-95, 2021. Google Scholar
  31. 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.
  32. Christian Konrad and Kheeran K. Naidu. On two-pass streaming algorithms for maximum bipartite matching. In Mary Wootters and Laura Sanità, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2021, August 16-18, 2021, University of Washington, Seattle, Washington, USA (Virtual Conference), volume 207 of LIPIcs, pages 19:1-19:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.19.
  33. Andrew McGregor. Graph stream algorithms: a survey. SIGMOD Rec., 43(1):9-20, 2014. URL: https://doi.org/10.1145/2627692.2627694.
  34. Rajeev Motwani and Prabhakar Raghavan. Randomized Algorithms. Cambridge University Press, 1995. Google Scholar
  35. S. Muthukrishnan. Data streams: Algorithms and applications. Found. Trends Theor. Comput. Sci., 1(2), 2005. URL: https://doi.org/10.1561/0400000002.
  36. Jelani Nelson and Huacheng Yu. Optimal lower bounds for distributed and streaming spanning forest computation. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1844-1860. SIAM, 2019. Google Scholar
  37. Eric Price and David P Woodruff. Lower bounds for adaptive sparse recovery. In Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms, pages 652-663. SIAM, 2013. Google Scholar
  38. Jeanette P. Schmidt, Alan Siegel, and Aravind Srinivasan. Chernoff-hoeffding bounds for applications with limited independence. SIAM J. Discret. Math., 8(2):223-250, 1995. Google Scholar
  39. 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 fuer Informatik, 2015. Google Scholar
  40. David P Woodruff and Qin Zhang. Tight bounds for distributed functional monitoring. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 941-960, 2012. Google Scholar
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