Maximum Weight b-Matchings in Random-Order Streams

Authors Chien-Chung Huang, François Sellier



PDF
Thumbnail PDF

File

LIPIcs.ESA.2022.68.pdf
  • Filesize: 0.7 MB
  • 14 pages

Document Identifiers

Author Details

Chien-Chung Huang
  • CNRS, DI ENS, École normale supérieure, Université PSL, Paris, France
François Sellier
  • Université Paris Cité, CNRS, IRIF, F-75013, Paris, France
  • MINES ParisTech, Université PSL, F-75006, Paris, France

Acknowledgements

The authors thank Claire Mathieu and the anonymous reviewers for their helpful comments.

Cite AsGet BibTex

Chien-Chung Huang and François Sellier. Maximum Weight b-Matchings in Random-Order Streams. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 68:1-68:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.68

Abstract

We consider the maximum weight b-matching problem in the random-order semi-streaming model. Assuming all weights are small integers drawn from [1,W], we present a 2 - 1/(2W) + ε approximation algorithm, using a memory of O(max(|M_G|, n) ⋅ poly(log(m),W,1/ε)), where |M_G| denotes the cardinality of the optimal matching. Our result generalizes that of Bernstein [Aaron Bernstein, 2020], which achieves a 3/2 + ε approximation for the maximum cardinality simple matching. When W is small, our result also improves upon that of Gamlath et al. [Gamlath et al., 2019], which obtains a 2 - δ approximation (for some small constant δ ∼ 10^{-17}) for the maximum weight simple matching. In particular, for the weighted b-matching problem, ours is the first result beating the approximation ratio of 2. Our technique hinges on a generalized weighted version of edge-degree constrained subgraphs, originally developed by Bernstein and Stein [Aaron Bernstein and Cliff Stein, 2015]. Such a subgraph has bounded vertex degree (hence uses only a small number of edges), and can be easily computed. The fact that it contains a 2 - 1/(2W) + ε approximation of the maximum weight matching is proved using the classical Kőnig-Egerváry’s duality theorem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Maximum weight matching
  • b-matching
  • streaming
  • random order

Metrics

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

References

  1. Sepehr Assadi and Soheil Behnezhad. Beating two-thirds for random-order streaming matching. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 19:1-19:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.19.
  2. Sepehr Assadi and Aaron Bernstein. Towards a unified theory of sparsification for matching problems. In Jeremy T. Fineman and Michael Mitzenmacher, editors, 2nd Symposium on Simplicity in Algorithms, SOSA 2019, January 8-9, 2019, San Diego, CA, USA, volume 69 of OASICS, pages 11:1-11:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/OASIcs.SOSA.2019.11.
  3. 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.
  4. Aaron Bernstein, Aditi Dudeja, and Zachary Langley. A framework for dynamic matching in weighted graphs. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 668-681. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451113.
  5. Aaron Bernstein and Cliff Stein. Fully dynamic matching in bipartite graphs. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, volume 9134 of Lecture Notes in Computer Science, pages 167-179. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-47672-7_14.
  6. Aaron Bernstein and Cliff Stein. Faster fully dynamic matchings with small approximation ratios. 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 692-711. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch50.
  7. Jeno Egerváry. Matrixok kombinatorius tulajdonságairól. Matematikai és Fizikai Lapok, 38(1931):16-28, 1931. Google Scholar
  8. Paul Erdős and László Lovász. Problems and results on 3-chromatic hypergraphs and some related questions. In Colloquia Mathematica Societatis Janos Bolyai 10. Infinite and Finite Sets, Keszthely (Hungary). Citeseer, 1973. Google Scholar
  9. Alireza Farhadi, Mohammad Taghi Hajiaghayi, Tung Mai, Anup Rao, and Ryan A. Rossi. Approximate maximum matching in random 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 1773-1785. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.108.
  10. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. Theoretical Computer Science, 348:207-216, December 2005. URL: https://doi.org/10.1016/j.tcs.2005.09.013.
  11. 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, pages 491-500, New York, NY, USA, 2019. Association for Computing Machinery. URL: https://doi.org/10.1145/3293611.3331603.
  12. Mohsen Ghaffari and David Wajc. Simplified and Space-Optimal Semi-Streaming (2+ε)-Approximate Matching. In Jeremy T. Fineman and Michael Mitzenmacher, editors, 2nd Symposium on Simplicity in Algorithms (SOSA 2019), volume 69 of OpenAccess Series in Informatics (OASIcs), pages 13:1-13:8, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/OASIcs.SOSA.2019.13.
  13. Guru Prashanth Guruganesh and Sahil Singla. Online matroid intersection: Beating half for random arrival. In Friedrich Eisenbrand and Jochen Könemann, editors, Integer Programming and Combinatorial Optimization - 19th International Conference, IPCO 2017, Waterloo, ON, Canada, June 26-28, 2017, Proceedings, volume 10328 of Lecture Notes in Computer Science, pages 241-253. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-59250-3_20.
  14. Chien-Chung Huang and François Sellier. Semi-streaming algorithms for submodular function maximization under b-matching constraint. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021), 2021. Google Scholar
  15. Michael Kapralov. Space lower bounds for approximating maximum matching in the edge arrival model. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10-13, 2021, pages 1874-1893. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.112.
  16. 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.
  17. 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.
  18. Roie Levin and David Wajc. Streaming submodular matching meets the primal-dual method. In SODA, pages 1914-1933, 2021. Google Scholar
  19. Ami Paz and Gregory Schwartzman. A (2 + ε)-approximation for maximum weight matching in the semi-streaming model. In Proceedings of the 2017 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2153-2161, 2017. URL: https://doi.org/10.1137/1.9781611974782.140.
  20. Moshe Tennenholtz. Some tractable combinatorial auctions. In AAAI/IAAI, pages 98-103, 2000. 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