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Improved Bounds for Matching in Random-Order Streams

Author Aaron Bernstein

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Author Details

Aaron Bernstein
  • Rutgers University, Department of Computer Science, New Brunswick, NJ, USA

Funding

  • Bernstein, Aaron: NSF CAREER Grant 1942010 and Simons Collaboration on Algorithms and Geometry.

Acknowledgements

I want to thank Sepehr Assadi for several very helpful discussions.

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Aaron Bernstein. Improved Bounds for Matching in Random-Order Streams. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 12:1-12:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.12

Abstract

We study the problem of computing an approximate maximum cardinality matching in the semi-streaming model when edges arrive in a random order. In the semi-streaming model, the edges of the input graph G = (V,E) are given as a stream e₁, …, e_m, and the algorithm is allowed to make a single pass over this stream while using O(n polylog(n)) space (m = |E| and n = |V|). If the order of edges is adversarial, a simple single-pass greedy algorithm yields a 1/2-approximation in O(n) space; achieving a better approximation in adversarial streams remains an elusive open question. A line of recent work shows that one can improve upon the 1/2-approximation if the edges of the stream arrive in a random order. The state of the art for this model is two-fold: Assadi et al. [SODA 2019] show how to compute a 2/3(∼.66)-approximate matching, but the space requirement is O(n^1.5 polylog(n)). Very recently, Farhadi et al. [SODA 2020] presented an algorithm with the desired space usage of O(n polylog(n)), but a worse approximation ratio of 6/11(∼.545), or 3/5(=.6) in bipartite graphs. In this paper, we present an algorithm that computes a 2/3(∼.66)-approximate matching using only O(n log(n)) space, improving upon both results above. We also note that for adversarial streams, a lower bound of Kapralov [SODA 2013] shows that any algorithm that achieves a 1-1/e(∼.63)-approximation requires (n^{1+Ω(1/log log(n))}) space. Our result for random-order streams is the first to go beyond the adversarial-order lower bound, thus establishing that computing a maximum matching is provably easier in random-order streams.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Graph Algorithms
  • Sublinear Algorithms
  • Matching
  • Streaming

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References

  1. Kook Jin Ahn and Sudipto Guha. Linear programming in the semi-streaming model with application to the maximum matching problem. Inf. Comput., 222:59-79, 2013. Google Scholar
  2. Kook Jin Ahn and Sudipto Guha. Access to data and number of iterations: Dual primal algorithms for maximum matching under resource constraints. In Proceedings of the 27th ACM on Symposium on Parallelism in Algorithms and Architectures, SPAA 2015, Portland, OR, USA, June 13-15, 2015, pages 202-211, 2015. Google Scholar
  3. 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 Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1616-1635, 2019. Google Scholar
  4. Sepehr Assadi and Aaron Bernstein. Towards a unified theory of sparsification for matching problems. In 2nd Symposium on Simplicity in Algorithms, SOSA@SODA 2019, January 8-9, 2019 - San Diego, CA, USA, pages 11:1-11:20, 2019. Google Scholar
  5. Sepehr Assadi, Sanjeev Khanna, and Yang Li. On estimating maximum matching size in graph streams. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 1723-1742, 2017. Google Scholar
  6. 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. Google Scholar
  7. Aaron Bernstein and Cliff Stein. Fully dynamic matching in bipartite graphs. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 167-179, 2015. Google Scholar
  8. Marc Bury and Chris Schwiegelshohn. Sublinear estimation of weighted matchings in dynamic data streams. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, pages 263-274, 2015. Google Scholar
  9. 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 Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1326-1344, 2016. Google Scholar
  10. Rajesh Hemant Chitnis, Graham Cormode, Mohammad Taghi Hajiaghayi, and Morteza Monemizadeh. Parameterized streaming: Maximal matching and vertex cover. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1234-1251, 2015. Google Scholar
  11. Graham Cormode, Hossein Jowhari, Morteza Monemizadeh, and S. Muthukrishnan. The sparse awakens: Streaming algorithms for matching size estimation in sparse graphs. In 25th Annual European Symposium on Algorithms, ESA 2017, September 4-6, 2017, Vienna, Austria, pages 29:1-29:15, 2017. Google Scholar
  12. Michael Crouch and Daniel S. Stubbs. Improved streaming algorithms for weighted matching, via unweighted matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2014, September 4-6, 2014, Barcelona, Spain, pages 96-104, 2014. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.96.
  13. Leah Epstein, Asaf Levin, Julián Mestre, and Danny Segev. Improved approximation guarantees for weighted matching in the semi-streaming model. SIAM J. Discrete Math., 25(3):1251-1265, 2011. Google Scholar
  14. Hossein Esfandiari, MohammadTaghi Hajiaghayi, Vahid Liaghat, Morteza Monemizadeh, and Krzysztof Onak. Streaming algorithms for estimating the matching size in planar graphs and beyond. ACM Trans. Algorithms, 14(4):48:1-48:23, 2018. Google Scholar
  15. Hossein Esfandiari, MohammadTaghi Hajiaghayi, and Morteza Monemizadeh. Finding large matchings in semi-streaming. In Carlotta Domeniconi, Francesco Gullo, Francesco Bonchi, Josep Domingo-Ferrer, Ricardo Baeza-Yates, Zhi-Hua Zhou, and Xindong Wu, editors, IEEE International Conference on Data Mining Workshops, ICDM Workshops 2016, December 12-15, 2016, Barcelona, Spain, pages 608-614. IEEE Computer Society, 2016. Google Scholar
  16. Alireza Farhadi, Mohammad Taghi Hajiaghayi, Tung Mai, Anup Rao, and Ryan A. Rossi. Approximate maximum matching in random streams. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1773-1785, 2020. Google Scholar
  17. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. Theor. Comput. Sci., 348(2-3):207-216, 2005. URL: https://doi.org/10.1016/j.tcs.2005.09.013.
  18. 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. Google Scholar
  19. Mohsen Ghaffari and David Wajc. Simplified and space-optimal semi-streaming (2+epsilon)-approximate matching. In Jeremy T. Fineman and Michael Mitzenmacher, editors, 2nd Symposium on Simplicity in Algorithms, SOSA@SODA 2019, January 8-9, 2019 - San Diego, CA, USA, volume 69 of OASICS, pages 13:1-13:8. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019. Google Scholar
  20. Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching. In Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '12, pages 468-485. SIAM, 2012. URL: http://dl.acm.org/citation.cfm?id=2095116.2095157.
  21. Venkatesan Guruswami and Krzysztof Onak. Superlinear lower bounds for multipass graph processing. In Proceedings of the 28th Conference on Computational Complexity, CCC 2013, K.lo Alto, California, USA, 5-7 June, 2013, pages 287-298, 2013. Google Scholar
  22. 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. Google Scholar
  23. Michael Kapralov. Better bounds for matchings in the streaming model. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 1679-1697, 2013. URL: https://doi.org/10.1137/1.9781611973105.121.
  24. Michael Kapralov, Sanjeev Khanna, and Madhu Sudan. Approximating matching size from random streams. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 734-751, 2014. URL: https://doi.org/10.1137/1.9781611973402.55.
  25. Michael Kapralov, Slobodan Mitrovic, Ashkan Norouzi-Fard, and Jakab Tardos. Space efficient approximation to maximum matching size from uniform edge samples. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1753-1772, 2020. Google Scholar
  26. Christian Konrad. Maximum matching in turnstile streams. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, pages 840-852, 2015. Google Scholar
  27. 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 fuer Informatik, 2018. Google Scholar
  28. Christian Konrad, Frédéric Magniez, and Claire Mathieu. Maximum matching in semi-streaming with few passes. In 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, pages 231-242, 2012. Google Scholar
  29. Andrew McGregor. Finding graph matchings in data streams. In Approximation, Randomization and Combinatorial Optimization, Algorithms and Techniques, 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2005 and 9th InternationalWorkshop on Randomization and Computation, RANDOM 2005, Berkeley, CA, USA, August 22-24, 2005, Proceedings, pages 170-181, 2005. URL: https://doi.org/10.1007/11538462_15.
  30. Andrew McGregor and Sofya Vorotnikova. Planar matching in streams revisited. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2016, September 7-9, 2016, Paris, France, pages 17:1-17:12, 2016. Google Scholar
  31. Andrew McGregor and Sofya Vorotnikova. A simple, space-efficient, streaming algorithm for matchings in low arboricity graphs. In 1st Symposium on Simplicity in Algorithms, SOSA 2018, January 7-10, 2018, New Orleans, LA, USA, pages 14:1-14:4, 2018. Google Scholar
  32. Ami Paz and Gregory Schwartzman. A (2 + epsilon)-approximation for maximum weight matching in the semi-streaming model. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2153-2161, 2017. Google Scholar
  33. David Wajc. Negative association: definition, properties, and applications. URL: https://www.cs.cmu.edu/~dwajc/notes/Negative%20Association.pdf.
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