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Constructing Large Matchings via Query Access to a Maximal Matching Oracle

Authors Lidiya Khalidah binti Khalil, Christian Konrad

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Lidiya Khalidah binti Khalil
  • Department of Computer Science, University of Bristol, UK
Christian Konrad
  • Department of Computer Science, University of Bristol, UK

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Lidiya Khalidah binti Khalil and Christian Konrad. Constructing Large Matchings via Query Access to a Maximal Matching Oracle. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.FSTTCS.2020.26

Abstract

Multi-pass streaming algorithm for Maximum Matching have been studied since more than 15 years and various algorithmic results are known today, including 2-pass streaming algorithms that break the 1/2-approximation barrier, and (1-ε)-approximation streaming algorithms that run in O(poly 1/ε) passes in bipartite graphs and in O((1/ε)^(1/ε)) or O(poly (1/ε) ⋅ log n) passes in general graphs, where n is the number of vertices of the input graph. However, proving impossibility results for such algorithms has so far been elusive, and, for example, even the existence of 2-pass small space streaming algorithms with approximation factor 0.999 has not yet been ruled out. The key building block of all multi-pass streaming algorithms for Maximum Matching is the Greedy matching algorithm. Our aim is to understand the limitations of this approach: How many passes are required if the algorithm solely relies on the invocation of the Greedy algorithm? In this paper, we initiate the study of lower bounds for restricted families of multi-pass streaming algorithms for Maximum Matching. We focus on the simple yet powerful class of algorithms that in each pass run Greedy on a vertex-induced subgraph of the input graph. In bipartite graphs, we show that 3 passes are necessary and sufficient to improve on the trivial approximation factor of 1/2: We give a lower bound of 0.6 on the approximation ratio of such algorithms, which is optimal. We further show that Ω(1/ε) passes are required for computing a (1-ε)-approximation, even in bipartite graphs. Last, the considered class of algorithms is not well-suited to general graphs: We show that Ω(n) passes are required in order to improve on the trivial approximation factor of 1/2.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Maximum matching approximation
  • Query model
  • Streaming algorithms

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References

  1. Hasan Abasi and Nader H. Bshouty. On learning graphs with edge-detecting queries. In Aurélien Garivier and Satyen Kale, editors, Algorithmic Learning Theory, ALT 2019, 22-24 March 2019, Chicago, Illinois, USA, volume 98 of Proceedings of Machine Learning Research, pages 3-30. PMLR, 2019. URL: http://proceedings.mlr.press/v98/abasi19a.html.
  2. Kook Jin Ahn and Sudipto Guha. Linear programming in the semi-streaming model with application to the maximum matching problem. Information and Computation, 222:59-79, 2013. 38th International Colloquium on Automata, Languages and Programming (ICALP 2011). URL: https://doi.org/10.1016/j.ic.2012.10.006.
  3. Noga Alon, Richard Beigel, Simon Kasif, Steven Rudich, and Benny Sudakov. Learning a hidden matching. SIAM J. Comput., 33(2):487-501, 2004. URL: https://doi.org/10.1137/S0097539702420139.
  4. 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.
  5. Sepehr Assadi, Deeparnab Chakrabarty, and Sanjeev Khanna. Graph connectivity and single element recovery via linear and or queries, 2020. URL: http://arxiv.org/abs/2007.06098.
  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. URL: https://doi.org/10.1137/1.9781611974331.ch93.
  7. Paul Beame, Sariel Har-Peled, Sivaramakrishnan Natarajan Ramamoorthy, Cyrus Rashtchian, and Makrand Sinha. Edge Estimation with Independent Set Oracles. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018), volume 94 of Leibniz International Proceedings in Informatics (LIPIcs), pages 38:1-38:21, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.38.
  8. 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), volume 168 of Leibniz International Proceedings in Informatics (LIPIcs), pages 12:1-12:13, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.12.
  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 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.
  10. Sung-Soon Choi. Polynomial time optimal query algorithms for finding graphs with arbitrary real weights. In Shai Shalev-Shwartz and Ingo Steinwart, editors, COLT 2013 - The 26th Annual Conference on Learning Theory, June 12-14, 2013, Princeton University, NJ, USA, volume 30 of JMLR Workshop and Conference Proceedings, pages 797-818. JMLR.org, 2013. URL: http://proceedings.mlr.press/v30/Choi13.html.
  11. Sung-Soon Choi and Jeong Han Kim. Optimal query complexity bounds for finding graphs. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC ’08, page 749–758, New York, NY, USA, 2008. Association for Computing Machinery. URL: https://doi.org/10.1145/1374376.1374484.
  12. Jacques Dark and Christian Konrad. Optimal lower bounds for matching and vertex cover in dynamic graph streams. In 35th Computational Complexity Conference, CCC 2020, LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  13. Sebastian Eggert, Lasse Kliemann, Peter Munstermann, and Anand Srivastav. Bipartite matching in the semi-streaming model. Algorithmica, 63(1–2):490–508, June 2012. Google Scholar
  14. 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. URL: https://doi.org/10.1109/ICDMW.2016.0092.
  15. 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
  16. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. Theor. Comput. Sci., 348(2):207–216, December 2005. URL: https://doi.org/10.1016/j.tcs.2005.09.013.
  17. Buddhima Gamlath, Sagar Kale, Slobodan Mitrovic, and Ola Svensson. Weighted matchings via unweighted augmentations. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, PODC ’19, page 491–500, New York, NY, USA, 2019. Association for Computing Machinery. URL: https://doi.org/10.1145/3293611.3331603.
  18. 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.
  19. Vladimir Grebinski and Gregory Kucherov. Optimal reconstruction of graphs under the additive model. Algorithmica, 28(1):104-124, 2000. URL: https://doi.org/10.1007/s004530010033.
  20. Venkatesan Guruswami and Krzysztof Onak. Superlinear lower bounds for multipass graph processing. Algorithmica, 76(3):654-683, 2016. URL: https://doi.org/10.1007/s00453-016-0138-7.
  21. 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.
  22. Michael Kapralov. Better bounds for matchings in the streaming model. In Sanjeev Khanna, editor, 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. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.121.
  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 Spirakis, and James Worrell, editors, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), volume 117 of Leibniz International Proceedings in Informatics (LIPIcs), pages 74:1-74:16, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.MFCS.2018.74.
  25. 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.
  26. Andrew McGregor. Finding graph matchings in data streams. In Proceedings of the 8th International Workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th International Conference on Randamization and Computation: Algorithms and Techniques, APPROX’05/RANDOM’05, page 170–181, Berlin, Heidelberg, 2005. Springer-Verlag. URL: https://doi.org/10.1007/11538462_15.
  27. Aviad Rubinstein, Tselil Schramm, and S. Matthew Weinberg. Computing Exact Minimum Cuts Without Knowing the Graph. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018), volume 94 of Leibniz International Proceedings in Informatics (LIPIcs), pages 39:1-39:16, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.39.
  28. Xiaoming Sun and David P. Woodruff. Tight Bounds for Graph Problems in Insertion Streams. In Naveen Garg, Klaus Jansen, Anup Rao, and José D. P. Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015), volume 40 of Leibniz International Proceedings in Informatics (LIPIcs), pages 435-448, Dagstuhl, Germany, 2015. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.435.
  29. Sumedh Tirodkar. Deterministic Algorithms for Maximum Matching on General Graphs in the Semi-Streaming Model. In Sumit Ganguly and Paritosh Pandya, editors, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018), volume 122 of Leibniz International Proceedings in Informatics (LIPIcs), pages 39:1-39:16, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2018.39.
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