Document Open Access Logo

A Simple Augmentation Method for Matchings with Applications to Streaming Algorithms

Author Christian Konrad



PDF
Thumbnail PDF

File

LIPIcs.MFCS.2018.74.pdf
  • Filesize: 492 kB
  • 16 pages

Document Identifiers

Author Details

Christian Konrad
  • Department of Computer Science, University of Bristol, Merchant Venturers Building, Woodland Road, BS8 1UB, United Kingdom

Cite AsGet BibTex

Christian Konrad. A Simple Augmentation Method for Matchings with Applications to Streaming Algorithms. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 74:1-74:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.MFCS.2018.74

Abstract

Given a graph G, it is well known that any maximal matching M in G is at least half the size of a maximum matching M^*. In this paper, we show that if G is bipartite, then running the Greedy matching algorithm on a sampled subgraph of G produces enough additional edges that can be used to augment M such that the resulting matching is of size at least (2 - sqrt{2})|M^*| ~~ 0.5857 |M^*| (ignoring lower order terms) with high probability. The main applications of our method lie in the area of data streaming algorithms, where an algorithm performs few passes over the edges of an n-vertex graph while maintaining a memory of size O(n polylog n). Our method immediately yields a very simple two-pass algorithm for Maximum Bipartite Matching (MBM) with approximation factor 0.5857, which only runs the Greedy matching algorithm in each pass. This slightly improves on the much more involved 0.583-approximation algorithm of Esfandiari et al. [ICDMW 2016]. To obtain our main result, we combine our method with a residual sparsity property of the random order Greedy algorithm and give a one-pass random order streaming algorithm for MBM with approximation factor 0.5395. This substantially improves upon the one-pass random order 0.505-approximation algorithm of Konrad et al. [APPROX 2012].

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Matchings
  • augmenting paths
  • streaming algorithms
  • random order

Metrics

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

References

  1. Kook Jin Ahn, Graham Cormode, Sudipto Guha, Andrew McGregor, and Anthony Wirth. Correlation clustering in data streams. In Proceedings of the 32nd International Conference on Machine Learning, ICML 2015, Lille, France, 6-11 July 2015, pages 2237-2246, 2015. URL: http://jmlr.org/proceedings/papers/v37/ahn15.html.
  2. 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. URL: http://dx.doi.org/10.1016/j.ic.2012.10.006.
  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. CoRR, abs/1711.03076, 2017. URL: http://arxiv.org/abs/1711.03076.
  4. Sepelir Assadi, Sanjeev Khanna, Yang Li, and Grigory Yaroslavtsev. Maximum matchings in dynamic graph streams and the simultaneous communication model. In Proceedings of the Twenty-seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '16, pages 1345-1364, Philadelphia, PA, USA, 2016. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2884435.2884528.
  5. Kazuoki Azuma. Weighted sums of certain dependent random variables. Tohoku Math. J. (2), 19(3):357-367, 1967. URL: http://dx.doi.org/10.2748/tmj/1178243286.
  6. 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. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_23.
  7. 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 '16, pages 1326-1344, Philadelphia, PA, USA, 2016. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2884435.2884527.
  8. Graham Cormode, Hossein Jowhari, Morteza Monemizadeh, and S. Muthukrishnan. The Sparse Awakens: Streaming Algorithms for Matching Size Estimation in Sparse Graphs. In Kirk Pruhs and Christian Sohler, editors, 25th Annual European Symposium on Algorithms (ESA 2017), volume 87 of Leibniz International Proceedings in Informatics (LIPIcs), pages 29:1-29:15, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.ESA.2017.29.
  9. Michael 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), volume 28 of Leibniz International Proceedings in Informatics (LIPIcs), pages 96-104, Dagstuhl, Germany, 2014. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.96.
  10. Michael S. Crouch, Andrew McGregor, and Daniel Stubbs. Dynamic graphs in the sliding-window model. In Hans L. Bodlaender and Giuseppe F. Italiano, editors, Algorithms - ESA 2013, pages 337-348, Berlin, Heidelberg, 2013. Springer Berlin Heidelberg. Google Scholar
  11. Jack Edmonds. Paths, trees and flowers. Canadian Journal of Mathematics, pages 449-467, 1965. Google Scholar
  12. Sebastian Eggert, Lasse Kliemann, Peter Munstermann, and Anand Srivastav. Bipartite matching in the semi-streaming model. Algorithmica, 63(1):490-508, Jun 2012. URL: http://dx.doi.org/10.1007/s00453-011-9556-8.
  13. 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 Proceedings of the Twenty-sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '15, pages 1217-1233, Philadelphia, PA, USA, 2015. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2722129.2722210.
  14. Hossein Esfandiari, MohammadTaghi Hajiaghayi, and Morteza Monemizadeh. Finding large matchings in semi-streaming. In IEEE International Conference on Data Mining Workshops, ICDM Workshops 2016, December 12-15, 2016, Barcelona, Spain., pages 608-614, 2016. URL: http://dx.doi.org/10.1109/ICDMW.2016.0092.
  15. Alexander Fanghänel, Thomas Kesselheim, and Berthold Vöcking. Improved algorithms for latency minimization in wireless networks. Theor. Comput. Sci., 412(24):2657-2667, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2010.05.004.
  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, 2005. URL: http://dx.doi.org/10.1016/j.tcs.2005.09.013.
  17. Mohsen Ghaffari, Themis Gouleakis, Slobodan Mitrovic, and Ronitt Rubinfeld. Improved massively parallel computation algorithms for mis, matching, and vertex cover. CoRR, abs/1802.08237, 2018. URL: http://arxiv.org/abs/1802.08237.
  18. 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 2012, Kyoto, Japan, January 17-19, 2012, pages 468-485, 2012. URL: http://portal.acm.org/citation.cfm?id=2095157&CFID=63838676&CFTOKEN=79617016.
  19. Venkatesan Guruswami and Krzysztof Onak. Superlinear lower bounds for multipass graph processing. Algorithmica, 76(3):654-683, 2016. URL: http://dx.doi.org/10.1007/s00453-016-0138-7.
  20. John E. Hopcroft and Richard M. Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing, 2(4):225-231, 1973. URL: http://dx.doi.org/10.1137/0202019.
  21. 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.
  22. Michael Kapralov. Better bounds for matchings in the streaming model. In Proceedings of the Twenty-fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '13, pages 1679-1697, Philadelphia, PA, USA, 2013. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2627817.2627938.
  23. 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 '14, pages 734-751, Philadelphia, PA, USA, 2014. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2634074.2634129.
  24. Christian Konrad. Maximum matching in turnstile streams. In Nikhil Bansal and Irene Finocchi, editors, Algorithms - ESA 2015, pages 840-852, Berlin, Heidelberg, 2015. Springer Berlin Heidelberg. Google Scholar
  25. Christian Konrad. MIS in the congested clique model in O(log log Δ) rounds. CoRR, abs/1802.07647, 2018. URL: http://arxiv.org/abs/1802.07647.
  26. 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. URL: http://dx.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: http://dx.doi.org/10.1145/2898960.
  28. Colin McDiarmid. On the method of bounded differences. In Surveys in Combinatorics 1989. Cambridge University Press, Cambridge, 1989. Google Scholar
  29. Andrew McGregor. Finding graph matchings in data streams. In Chandra Chekuri, Klaus Jansen, José D. P. Rolim, and Luca Trevisan, editors, Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, pages 170-181, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg. Google Scholar
  30. Andrew McGregor and Sofya Vorotnikova. A Simple, Space-Efficient, Streaming Algorithm for Matchings in Low Arboricity Graphs. In Raimund Seidel, editor, 1st Symposium on Simplicity in Algorithms (SOSA 2018), volume 61 of OpenAccess Series in Informatics (OASIcs), pages 14:1-14:4, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/OASIcs.SOSA.2018.14.
  31. Ami Paz and Gregory Schwartzman. A 2 + ε-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. URL: http://dx.doi.org/10.1137/1.9781611974782.140.
  32. 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: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.435.
  33. Mariano Zelke. Weighted matching in the semi-streaming model. Algorithmica, 62(1):1-20, Feb 2012. URL: http://dx.doi.org/10.1007/s00453-010-9438-5.
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