Document Open Access Logo

Streaming Submodular Maximization Under Matroid Constraints

Authors Moran Feldman , Paul Liu , Ashkan Norouzi-Fard , Ola Svensson , Rico Zenklusen

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2022.59.pdf
  • Filesize: 1.13 MB
  • 20 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Submodular optimization and polymatroids
  • Theory of computation → Streaming models
Keywords
  • Submodular maximization
  • streaming
  • matroid
  • random order

Metrics

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

Abstract

Recent progress in (semi-)streaming algorithms for monotone submodular function maximization has led to tight results for a simple cardinality constraint. However, current techniques fail to give a similar understanding for natural generalizations, including matroid constraints. This paper aims at closing this gap. For a single matroid of rank k (i.e., any solution has cardinality at most k), our main results are:  
- A single-pass streaming algorithm that uses Õ(k) memory and achieves an approximation guarantee of 0.3178. 
- A multi-pass streaming algorithm that uses Õ(k) memory and achieves an approximation guarantee of (1-1/e - ε) by taking a constant (depending on ε) number of passes over the stream.  This improves on the previously best approximation guarantees of 1/4 and 1/2 for single-pass and multi-pass streaming algorithms, respectively. In fact, our multi-pass streaming algorithm is tight in that any algorithm with a better guarantee than 1/2 must make several passes through the stream and any algorithm that beats our guarantee of 1-1/e must make linearly many passes (as well as an exponential number of value oracle queries). 
Moreover, we show how the approach we use for multi-pass streaming can be further strengthened if the elements of the stream arrive in uniformly random order, implying an improved result for p-matchoid constraints.

Cite As Get BibTex

Moran Feldman, Paul Liu, Ashkan Norouzi-Fard, Ola Svensson, and Rico Zenklusen. Streaming Submodular Maximization Under Matroid Constraints. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 59:1-59:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.59

Author Details

Moran Feldman
  • University of Haifa, Israel
Paul Liu
  • Stanford University, CA, USA
Ashkan Norouzi-Fard
  • Google Research, Zurich, Switzerland
Ola Svensson
  • EPFL, Lausanne, Switzerland
Rico Zenklusen
  • ETH Zürich, Switzerland

Funding

  • Feldman, Moran: Research supported in part by the Israel Science Foundation (ISF) grants no. 1357/16 and 459/20.
  • Svensson, Ola: Research supported by the Swiss National Science Foundation project 200021-184656 "Randomness in Problem Instances and Randomized Algorithms."
  • Zenklusen, Rico: Research supported in part by Swiss National Science Foundation grant number 200021_184622. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 817750).

Acknowledgements

The authors are very grateful to Jan Vondrák. It is safe to say that this paper would not be nearly as good without Jan’s many interesting discussions and comments.

References

  1. Shipra Agrawal, Mohammad Shadravan, and Cliff Stein. Submodular secretary problem with shortlists. In Avrim Blum, editor, 10th Innovations in Theoretical Computer Science Conference (ITCS), pages 1:1-1:19, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.1.
  2. Ashwinkumar Badanidiyuru, Baharan Mirzasoleiman, Amin Karbasi, and Andreas Krause. Streaming submodular maximization: Massive data summarization on the fly. In Proc. of the 20th ACM Conference on Knowledge Discovery and Data Mining (KDD), pages 671-680, 2014. Google Scholar
  3. Ashwinkumar Badanidiyuru and Jan Vondrák. Fast algorithms for maximizing submodular functions. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SODA '14, pages 1497-1514, 2014. Google Scholar
  4. Niv Buchbinder and Moran Feldman. Constrained submodular maximization via a nonsymmetric technique. Mathematics of Operations Research, 44(3):988-1005, 2019. Google Scholar
  5. Gruia Călinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Comput., 40(6):1740-1766, 2011. URL: https://doi.org/10.1137/080733991.
  6. Amit Chakrabarti and Sagar Kale. Submodular maximization meets streaming: matchings, matroids, and more. Mathematical Programming, 154(1-2):225-247, 2015. Google Scholar
  7. T-H. Hubert Chan, Zhiyi Huang, Shaofeng H.-C. Jiang, Ning Kang, and Zhihao Gavin Tang. Online submodular maximization with free disposal: Randomization beats 1/4 for partition matroids. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1204-1223, 2017. Google Scholar
  8. Chandra Chekuri, Shalmoli Gupta, and Kent Quanrud. Streaming algorithms for submodular function maximization. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming, pages 318-330, Berlin, Heidelberg, 2015. Springer Berlin Heidelberg. Google Scholar
  9. Alina Ene and Huy L. Nguyen. Constrained submodular maximization: Beyond 1/e. In IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 248-257, 2016. Google Scholar
  10. Uriel Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634-652, 1998. Google Scholar
  11. M. Feldman, A. Norouzi-Fard, O. Svensson, and R. Zenklusen. The one-way communication complexity of submodular maximization with applications to streaming and robustness. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory on Computing (STOC), pages 1363-1374, 2020. Google Scholar
  12. Moran Feldman, Amin Karbasi, and Ehsan Kazemi. Do less, get more: Streaming submodular maximization with subsampling. In Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa-Bianchi, and Roman Garnett, editors, Advances in Neural Information Processing Systems (NeurIPS), pages 730-740, 2018. URL: https://proceedings.neurips.cc/paper/2018/hash/d1f255a373a3cef72e03aa9d980c7eca-Abstract.html.
  13. Moran Feldman, Joseph Naor, and Roy Schwartz. A unified continuous greedy algorithm for submodular maximization. In IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), pages 570-579, 2011. Google Scholar
  14. Yuval Filmus and Justin Ward. Monotone submodular maximization over a matroid via non-oblivious local search. SIAM Journal on Computing, 43(2):514-542, 2014. Google Scholar
  15. Shayan Oveis Gharan and Jan Vondrák. Submodular maximization by simulated annealing. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1098-1116, 2011. Google Scholar
  16. Ran Haba, Ehsan Kazemi, Moran Feldman, and Amin Karbasi. Streaming submodular maximization under a k-set system constraint. In Proceedings of the 37th International Conference on Machine Learning (ICML), pages 3939-3949, 2020. Google Scholar
  17. Chien-Chung Huang and Naonori Kakimura. Multi-pass streaming algorithms for monotone submodular function maximization. CoRR, abs/1802.06212, 2018. Google Scholar
  18. Chien-Chung Huang, Theophile Thiery, and Justin Ward. Improved multi-pass streaming algorithms for submodular maximization with matroid constraints, 2021. URL: http://arxiv.org/abs/2102.09679.
  19. Andreas Krause. Submodularity in machine learning. URL: http://submodularity.org/.
  20. Jon Lee, Vahab S. Mirrokni, Viswanath Nagarajan, and Maxim Sviridenko. Non-monotone submodular maximization under matroid and knapsack constraints. In Michael Mitzenmacher, editor, Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 323-332. ACM, 2009. URL: https://doi.org/10.1145/1536414.1536459.
  21. Paul Liu, Aviad Rubinstein, Jan Vondrák, and Junyao Zhao. Cardinality constrained submodular maximization for random streams. In Advances in Neural Information Processing Systems (NeurIPS), 2021. URL: http://arxiv.org/abs/2111.07217.
  22. Andrew McGregor and Hoa T. Vu. Better streaming algorithms for the maximum coverage problem. Theory of Computing Systems, 63(7):1595-1619, 2019. URL: https://doi.org/10.1007/s00224-018-9878-x.
  23. Baharan Mirzasoleiman, Ashwinkumar Badanidiyuru, and Amin Karbasi. Fast constrained submodular maximization: Personalized data summarization. In Proceedings of the 33nd International Conference on Machine Learning, ICML, pages 1358-1367, 2016. Google Scholar
  24. Baharan Mirzasoleiman, Stefanie Jegelka, and Andreas Krause. Streaming non-monotone submodular maximization: Personalized video summarization on the fly. In Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18), pages 1379-1386, 2018. Google Scholar
  25. George L. Nemhauser and Laurence A. Wolsey. Best algorithms for approximating the maximum of a submodular set function. Math. Oper. Res., 3(3):177-188, 1978. Google Scholar
  26. George L. Nemhauser, Laurence A. Wolsey, and Marshall L. Fisher. An analysis of approximations for maximizing submodular set functions - I. Math. Program., 14(1):265-294, 1978. Google Scholar
  27. Ashkan Norouzi-Fard, Jakub Tarnawski, Slobodan Mitrovic, Amir Zandieh, Aidasadat Mousavifar, and Ola Svensson. Beyond 1/2-approximation for submodular maximization on massive data streams. In Jennifer G. Dy and Andreas Krause, editors, Proceedings of the 35th International Conference on Machine Learning (ICML), volume 80 of Proceedings of Machine Learning Research, pages 3826-3835. PMLR, 2018. URL: http://proceedings.mlr.press/v80/norouzi-fard18a.html.
  28. Alexander Schrijver. Combinatorial Optimization : Polyhedra and Efficiency. Springer-Verlag Berlin Heidelberg, 2003. Google Scholar
  29. Mohammad Shadravan. Submodular Matroid Secretary Problem with Shortlists, January 2020. URL: http://arxiv.org/abs/2001.00894.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact