Document Open Access Logo

Beating the Folklore Algorithm for Dynamic Matching

Authors Mohammad Roghani, Amin Saberi, David Wajc

PDF

File

Thumbnail PDF
PDF
LIPIcs.ITCS.2022.111.pdf
  • Filesize: 0.81 MB
  • 23 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • dynamic matching
  • dynamic graph algorithms
  • sublinear algorithms

Metrics

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

Abstract

The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore algorithm, which maintains a maximal (and hence 2-approximate) matching in O(n) worst-case update time in n-node graphs. 

We present the first deterministic algorithm which outperforms the folklore algorithm in terms of both approximation ratio and worst-case update time. Specifically, we give a (2-Ω(1))-approximate algorithm with O(m^{3/8}) = O(n^{3/4}) worst-case update time in n-node, m-edge graphs. For sufficiently small constant ε > 0, no deterministic (2+ε)-approximate algorithm with worst-case update time O(n^{0.99}) was known. Our second result is the first deterministic (2+ε)-approximate weighted matching algorithm with O_ε(1)⋅ O(∜{m}) = O_ε(1)⋅ O(√n) worst-case update time. Neither of our results were previously known to be achievable by a randomized algorithm against an adaptive adversary.

Our main technical contributions are threefold: first, we characterize the tight cases for kernels, which are the well-studied matching sparsifiers underlying much of the (2+ε)-approximate dynamic matching literature. This characterization, together with multiple ideas - old and new - underlies our result for breaking the approximation barrier of 2. Our second technical contribution is the first example of a dynamic matching algorithm whose running time is improved due to improving the recourse of other dynamic matching algorithms. Finally, we show how to use dynamic bipartite matching algorithms as black-box subroutines for dynamic matching in general graphs without incurring the natural 3/2 factor in the approximation ratio which such approaches naturally incur (reminiscent of the integrality gap of the fractional matching polytope in general graphs).

Cite As Get BibTex

Mohammad Roghani, Amin Saberi, and David Wajc. Beating the Folklore Algorithm for Dynamic Matching. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 111:1-111:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.111

Author Details

Mohammad Roghani
  • Stanford University, CA, USA
  • roghani@stanford.edu
Amin Saberi
  • Stanford University, CA, USA
  • saberi@stanford.edu
David Wajc
  • Stanford University, CA, USA
  • wajc@stanford.edu

Funding

This research is partially supported by NSF award 1812919, ONR award N000141912550, and a gift from Cisco Research.

Acknowledgements

The authors thank Sayan Bhattacharya and Aaron Bernstein for helpful discussions, and Aaron Bernstein for sharing a preprint of [Bernstein et al., 2021].

References

  1. Moab Arar, Shiri Chechik, Sarel Cohen, Cliff Stein, and David Wajc. Dynamic matching: Reducing integral algorithms to approximately-maximal fractional algorithms. In Proceedings of the 45th International Colloquium on Automata, Languages and Programming (ICALP), pages 79:1-79:16, 2018. Google Scholar
  2. Surender Baswana, Manoj Gupta, and Sandeep Sen. Fully dynamic maximal matching in O(log n) update time. SIAM Journal on Computing (SICOMP), 44(1):88-113, 2015. Google Scholar
  3. Soheil Behnezhad, Mahsa Derakhshan, MohammadTaghi Hajiaghayi, Cliff Stein, and Madhu Sudan. Fully dynamic maximal independent set with polylogarithmic update time. In Proceedings of the 60th Symposium on Foundations of Computer Science (FOCS), pages 382-405, 2019. Google Scholar
  4. Soheil Behnezhad, Jakub Łącki, and Vahab Mirrokni. Fully dynamic matching: Beating 2-approximation in Δ^ε update time. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2492-2508, 2020. Google Scholar
  5. Aaron Bernstein, Aditi Dudeja, and Zachary Langley. A framework for dynamic matching in weighted graphs. In Proceedings of the 53rd Annual ACM Symposium on Theory of Computing (STOC), 2021. Google Scholar
  6. Aaron Bernstein, Sebastian Forster, and Monika Henzinger. A deamortization approach for dynamic spanner and dynamic maximal matching. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1899-1918, 2019. Google Scholar
  7. Aaron Bernstein and Cliff Stein. Fully dynamic matching in bipartite graphs. In Proceedings of the 42nd International Colloquium on Automata, Languages and Programming (ICALP), pages 167-179, 2015. Google Scholar
  8. Aaron Bernstein and Cliff Stein. Faster fully dynamic matchings with small approximation ratios. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 692-711, 2016. Google Scholar
  9. Sayan Bhattacharya, Deeparnab Chakrabarty, and Monika Henzinger. Deterministic dynamic matching in O(1) update time. Algorithmica, 82(4):1057-1080, 2020. Google Scholar
  10. Sayan Bhattacharya, Fabrizio Grandoni, and David Wajc. Online edge coloring algorithms via the nibble method. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2830-2842. SIAM, 2021. Google Scholar
  11. Sayan Bhattacharya, Monika Henzinger, and Giuseppe F Italiano. Deterministic fully dynamic data structures for vertex cover and matching. SIAM Journal on Computing (SICOMP), 47(3):859-887, 2018. Google Scholar
  12. Sayan Bhattacharya, Monika Henzinger, and Danupon Nanongkai. New deterministic approximation algorithms for fully dynamic matching. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing (STOC), pages 398-411, 2016. Google Scholar
  13. Sayan Bhattacharya and Peter Kiss. Deterministic rounding of dynamic fractional matchings. In Proceedings of the 48th International Colloquium on Automata, Languages and Programming (ICALP), 2021. Google Scholar
  14. Moses Charikar and Shay Solomon. Fully dynamic almost-maximal matching: Breaking the polynomial barrier for worst-case time bounds. In Proceedings of the 45th International Colloquium on Automata, Languages and Programming (ICALP), pages 33:1-33:14, 2018. Google Scholar
  15. Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. Introduction to algorithms. MIT press, 2009. Google Scholar
  16. Søren Dahlgaard. On the hardness of partially dynamic graph problems and connections to diameter. In Proceedings of the 43rd International Colloquium on Automata, Languages and Programming (ICALP), pages 48:1-48:14, 2016. Google Scholar
  17. Ran Duan and Seth Pettie. Linear-time approximation for maximum weight matching. Journal of the ACM (JACM), 61(1):1, 2014. Google Scholar
  18. Buddhima Gamlath, Michael Kapralov, Andreas Maggiori, Ola Svensson, and David Wajc. Online matching with general arrivals. In Proceedings of the 60th Symposium on Foundations of Computer Science (FOCS), pages 26-37, 2019. Google Scholar
  19. Fabrizio Grandoni, Chris Schwiegelshohn, Shay Solomon, and Amitai Uzrad. Maintaining an edcs in general graphs: Simpler, density-sensitive and with worst-case time bounds. To appear in 5th Symposium on Simplicity in Algorithms (SOSA), 2022. Google Scholar
  20. Anupam Gupta and Roie Levin. Fully-dynamic submodular cover with bounded recourse. In Proceedings of the 61st Symposium on Foundations of Computer Science (FOCS), pages 1147-1157, 2020. Google Scholar
  21. Manoj Gupta and Richard Peng. Fully dynamic (1+ε)-approximate matchings. In Proceedings of the 54th Symposium on Foundations of Computer Science (FOCS), pages 548-557, 2013. Google Scholar
  22. Kathrin Hanauer, Monika Henzinger, and Christian Schulz. Recent advances in fully dynamic graph algorithms. arXiv preprint arXiv:2102.11169, 2021. Google Scholar
  23. Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC), pages 21-30, 2015. Google Scholar
  24. Peter Kiss. Improving update times of dynamic matching algorithms from amortized to worst case. To appear in 13th Innovations in Theoretical Computer Science Conference (ITCS), 2022. Google Scholar
  25. Christian Konrad, Frédéric Magniez, and Claire Mathieu. Maximum matching in semi-streaming with few passes. In Proceedings of the 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pages 231-242, 2012. Google Scholar
  26. Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Higher lower bounds from the 3sum conjecture. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1272-1287, 2016. Google Scholar
  27. Silvio Micali and Vijay V Vazirani. An O(√|V|⋅|E|) algoithm for finding maximum matching in general graphs. In Proceedings of the 21st Symposium on Foundations of Computer Science (FOCS), pages 17-27, 1980. Google Scholar
  28. Danupon Nanongkai and Thatchaphol Saranurak. Dynamic spanning forest with worst-case update time: adaptive, las vegas, and O(n^1/2-ε)-time. In Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC), pages 1122-1129, 2017. Google Scholar
  29. Ofer Neiman and Shay Solomon. Simple deterministic algorithms for fully dynamic maximal matching. ACM Transactions on Algorithms (TALG), 12(1):7, 2016. Google Scholar
  30. Krzysztof Onak and Ronitt Rubinfeld. Maintaining a large matching and a small vertex cover. In Proceedings of the 42nd Annual ACM Symposium on Theory of Computing (STOC), pages 457-464, 2010. Google Scholar
  31. David Peleg and Shay Solomon. Dynamic (1+ ε)-approximate matchings: a density-sensitive approach. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 712-729, 2016. Google Scholar
  32. Noam Solomon and Shay Solomon. A generalized matching reconfiguration problem. In Proceedings of the 12th Innovations in Theoretical Computer Science Conference (ITCS), 2021. Google Scholar
  33. Shay Solomon. Fully dynamic maximal matching in constant update time. In Proceedings of the 57th Symposium on Foundations of Computer Science (FOCS), pages 325-334, 2016. Google Scholar
  34. Shay Solomon. Local algorithms for bounded degree sparsifiers in sparse graphs. In Proceedings of the 9th Innovations in Theoretical Computer Science Conference (ITCS), pages 52:1-52:19, 2018. Google Scholar
  35. David Wajc. Rounding dynamic matchings against an adaptive adversary. In Proceedings of the 52nd Annual ACM Symposium on Theory of Computing (STOC), pages 194-207, 2020. 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