Document Open Access Logo

Deterministic Algorithms for Maximum Matching on General Graphs in the Semi-Streaming Model

Author Sumedh Tirodkar

PDF

File

Thumbnail PDF
PDF
LIPIcs.FSTTCS.2018.39.pdf
  • Filesize: 487 kB
  • 16 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Semi Streaming
  • Maximum Matching

Metrics

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

Abstract

We present an improved deterministic algorithm for Maximum Cardinality Matching on general graphs in the Semi-Streaming Model. In the Semi-Streaming Model, a graph is presented as a sequence of edges, and an algorithm must access the edges in the given sequence. It can only use O(n polylog n) space to perform computations, where n is the number of vertices of the graph. If the algorithm goes over the stream k times, it is called a k-pass algorithm. In this model, McGregor [McGregor, 2005] gave the currently best known randomized (1+epsilon)-approximation algorithm for maximum cardinality matching on general graphs, that uses (1/epsilon)^{O(1/epsilon)} passes. Ahn and Guha [Ahn and Guha, 2013] later gave the currently best known deterministic (1+epsilon)-approximation algorithms for maximum cardinality matching: one on bipartite graphs that uses O(log log(1/epsilon)/epsilon^2) passes, and the other on general graphs that uses O(log n *poly(1/epsilon)) passes (note that, for general graphs, the number of passes is dependent on the size of the input). We present the first deterministic algorithm that achieves a (1+epsilon)-approximation on general graphs in only a constant number ((1/epsilon)^{O(1/epsilon)}) of passes.

Cite As Get BibTex

Sumedh Tirodkar. Deterministic Algorithms for Maximum Matching on General Graphs in the Semi-Streaming Model. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.FSTTCS.2018.39

Author Details

Sumedh Tirodkar
  • IDSIA, USI-SUPSI, Manno, Switzerland

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, January 2013. URL: http://dx.doi.org/10.1016/j.ic.2012.10.006.
  2. Sepehr Assadi, Sanjeev Khanna, and Yang Li. On Estimating Maximum Matching Size in Graph Streams. In Proc. 28th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1723-1742, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.113.
  3. Sepehr Assadi, Sanjeev Khanna, Yang Li, and Grigory Yaroslavtsev. Maximum Matchings in Dynamic Graph Streams and the Simultaneous Communication Model. In Proc. 27th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1345-1364, 2016. URL: http://dl.acm.org/citation.cfm?id=2884435.2884528.
  4. Aaron Bernstein and Cliff Stein. Faster Fully Dynamic Matchings with Small Approximation Ratios. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 692-711, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch50.
  5. Sayan Bhattacharya, Monika Henzinger, and Giuseppe F. Italiano. Deterministic Fully Dynamic Data Structures for Vertex Cover and Matching. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 785-804, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.54.
  6. Sayan Bhattacharya, Monika Henzinger, and Danupon Nanongkai. Fully Dynamic Approximate Maximum Matching and Minimum Vertex Cover in O(log^3 n) Worst Case Update Time. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 470-489, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.30.
  7. Niv Buchbinder, Danny Segev, and Yevgeny Tkach. Online Algorithms for Maximum Cardinality Matching with Edge Arrivals. In 25th Annual European Symposium on Algorithms, ESA 2017, September 4-6, 2017, Vienna, Austria, pages 22:1-22:14, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ESA.2017.22.
  8. Marc Bury and Chris Schwiegelshohn. Sublinear Estimation of Weighted Matchings in Dynamic Data Streams. In Proc. 23rd Annual European Symposium on Algorithms, pages 263-274, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_23.
  9. Amit Chakrabarti and Sagar Kale. Submodular maximization meets streaming: matchings, matroids, and more. Mathematical Programming, 154(1):225-247, 2015. URL: http://dx.doi.org/10.1007/s10107-015-0900-7.
  10. Chandra Chekuri, Shalmoli Gupta, and Kent Quanrud. Streaming Algorithms for Submodular Function Maximization. In Proc. 42nd International Colloquium on Automata, Languages and Programming, pages 318-330, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_26.
  11. Ashish Chiplunkar, Sumedh Tirodkar, and Sundar Vishwanathan. On Randomized Algorithms for Matching in the Online Preemptive Model. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, pages 325-336, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_28.
  12. 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 Proc. 27th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1326-1344, 2016. URL: http://dl.acm.org/citation.cfm?id=2884435.2884527.
  13. Sebastian Eggert, Lasse Kliemann, Peter Munstermann, and Anand Srivastav. Bipartite Matching in the Semi-streaming Model. Algorithmica, 63(1):490-508, 2012. URL: http://dx.doi.org/10.1007/s00453-011-9556-8.
  14. Leah Epstein, Asaf Levin, Julian Mestre, and Danny Segev. Improved Approximation Guarantees for Weighted Matching in the Semi-streaming Model. SIAM Journal on Discrete Mathematics, 25(3):1251-1265, 2011. URL: http://dx.doi.org/10.1137/100801901.
  15. Leah Epstein, Asaf Levin, Danny Segev, and Oren Weimann. Improved Bounds for Online Preemptive Matching. In 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, Kiel, Germany, pages 389-399, 2013. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2013.389.
  16. H. Esfandiari, M. Hajiaghayi, and M. Monemizadeh. Finding Large Matchings in Semi-Streaming. In 2016 IEEE 16th International Conference on Data Mining Workshops (ICDMW), pages 608-614, December 2016. URL: http://dx.doi.org/10.1109/ICDMW.2016.0092.
  17. Hossein Esfandiari, Mohammad T. Hajiaghayi, Vahid Liaghat, Morteza Monemizadeh, and Krzysztof Onak. Streaming Algorithms for Estimating the Matching Size in Planar Graphs and Beyond. In Proc. 26th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1217-1233, 2015. URL: http://dl.acm.org/citation.cfm?id=2722129.2722210.
  18. 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: http://dx.doi.org/10.1016/j.tcs.2005.09.013.
  19. Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching. In Proc. 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 468-485, 2012. URL: http://dl.acm.org/citation.cfm?id=2095116.2095157.
  20. Sagar Kale and Sumedh Tirodkar. Maximum Matching in Two, Three, and a Few More Passes Over Graph Streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2017, August 16-18, 2017, Berkeley, CA, USA, pages 15:1-15:21, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.15.
  21. Michael Kapralov. Better bounds for matchings in the streaming model. In Proc. 24th Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 2013. Google Scholar
  22. Michael Kapralov, Sanjeev Khanna, and Madhu Sudan. Approximating Matching Size from Random Streams. In Proc. 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 734-751, 2014. URL: http://dl.acm.org/citation.cfm?id=2634074.2634129.
  23. R. M. Karp, U. V. Vazirani, and V. V. Vazirani. An Optimal Algorithm for On-line Bipartite Matching. In Proceedings of the Twenty-second Annual ACM Symposium on Theory of Computing, STOC '90, pages 352-358, New York, NY, USA, 1990. ACM. Google Scholar
  24. Christian Konrad. Maximum Matching in Turnstile Streams. In Proc. 23rd Annual European Symposium on Algorithms, pages 840-852, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_70.
  25. Christian Konrad, Frédéric Magniez, and Claire Mathieu. Maximum Matching in Semi-Streaming with Few Passes. CoRR, abs/1112.0184, 2014. URL: http://arxiv.org/abs/1112.0184.
  26. Mohammad Mahdian and Qiqi Yan. Online Bipartite Matching with Random Arrivals: An Approach Based on Strongly Factor-revealing LPs. In Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing, STOC '11, pages 597-606, New York, NY, USA, 2011. ACM. URL: http://dx.doi.org/10.1145/1993636.1993716.
  27. Andrew McGregor. Problem 60: Single-Pass Unweighted Matchings. http://sublinear.info/index.php?title=Open_Problems:60. Accessed: 2018-02-10.
  28. Andrew McGregor. Finding graph matchings in data streams. In Proc. 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pages 170-181, 2005. URL: http://dx.doi.org/10.1007/11538462_15.
  29. Silvio Micali and Vijay V. Vazirani. An O(√|V||E|) Algorithm for Finding Maximum Matching in General Graphs. In Proceedings of the 21st Annual Symposium on Foundations of Computer Science, SFCS '80, pages 17-27, Washington, DC, USA, 1980. IEEE Computer Society. URL: http://dx.doi.org/10.1109/SFCS.1980.12.
  30. Ami Paz and Gregory Schwartzman. A (2 + ε)-Approximation for Maximum Weight Matching in the Semi-Streaming Model. In Proc. 28th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2153-2161, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.140.
  31. A. Schrijver. Combinatorial Optimization - Polyhedra and Efficiency. Springer, 2003. Google Scholar
  32. Shay Solomon. Fully Dynamic Maximal Matching in Constant Update Time. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 325-334, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.43.
  33. Mariano Zelke. Weighted matching in the semi-streaming model. In Proc. 25th International Symposium on Theoretical Aspects of Computer Science, pages 669-680, 2008. 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact