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Regularized Box-Simplex Games and Dynamic Decremental Bipartite Matching

Authors Arun Jambulapati, Yujia Jin, Aaron Sidford, Kevin Tian

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Author Details

Arun Jambulapati
  • Stanford University, CA, USA
Yujia Jin
  • Stanford University, CA, USA
Aaron Sidford
  • Stanford University, CA, USA
Kevin Tian
  • Stanford University, CA, USA

Funding

  • Jin, Yujia: supported by a Stanford Graduate Fellowship and the Dantzig-Lieberman Operations Research Fellowship.
  • Sidford, Aaron: supported in part by a Microsoft Research Faculty Fellowship, NSF CAREER Award CCF-1844855, NSF Grant CCF-1955039, a PayPal research award, and a Sloan Research Fellowship.
  • Tian, Kevin: supported in part by a Google Ph.D. Fellowship, a Simons-Berkeley VMware Research Fellowship, a Microsoft Research Faculty Fellowship, NSF CAREER Award CCF-1844855, NSF Grant CCF-1955039, and a PayPal research award.

Acknowledgements

We thank anonymous reviewers for their feedback, Amin Saberi and David Wajc for helpful conversations, Jason Altschuler for providing a reference for the unaccelerated convergence rate of Sinkhorn's algorithm (in the original submission, we claimed no such rate had been stated previously), and Monika Henzinger and Thatchaphol Saranurak for helpful information regarding prior work.

Cite As Get BibTex

Arun Jambulapati, Yujia Jin, Aaron Sidford, and Kevin Tian. Regularized Box-Simplex Games and Dynamic Decremental Bipartite Matching. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 77:1-77:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.77

Abstract

Box-simplex games are a family of bilinear minimax objectives which encapsulate graph-structured problems such as maximum flow [Sherman, 2017], optimal transport [Arun Jambulapati et al., 2019], and bipartite matching [Sepehr Assadi et al., 2022]. We develop efficient near-linear time, high-accuracy solvers for regularized variants of these games. Beyond the immediate applications of such solvers for computing Sinkhorn distances, a prominent tool in machine learning, we show that these solvers can be used to obtain improved running times for maintaining a (fractional) ε-approximate maximum matching in a dynamic decremental bipartite graph against an adaptive adversary. We give a generic framework which reduces this dynamic matching problem to solving regularized graph-structured optimization problems to high accuracy. Through our reduction framework, our regularized box-simplex game solver implies a new algorithm for dynamic decremental bipartite matching in total time Õ(m ⋅ ε^{-3}), from an initial graph with m edges and n nodes. We further show how to use recent advances in flow optimization [Chen et al., 2022] to improve our runtime to m^{1 + o(1)} ⋅ ε^{-2}, thereby demonstrating the versatility of our reduction-based approach. These results improve upon the previous best runtime of Õ(m ⋅ ε^{-4}) [Aaron Bernstein et al., 2020] and illustrate the utility of using regularized optimization problem solvers for designing dynamic algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • bipartite matching
  • decremental matching
  • dynamic algorithms
  • continuous optimization
  • box-simplex games
  • primal-dual method

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References

  1. Ittai Abraham, David Durfee, Ioannis Koutis, Sebastian Krinninger, and Richard Peng. On fully dynamic graph sparsifiers. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 335-344. IEEE Computer Society, 2016. Google Scholar
  2. Zeyuan Allen-Zhu, Yuanzhi Li, Rafael Mendes de Oliveira, and Avi Wigderson. Much faster algorithms for matrix scaling. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 890-901, 2017. Google Scholar
  3. Jason Altschuler, Francis Bach, Alessandro Rudi, and Jonathan Niles-Weed. Massively scalable sinkhorn distances via the nyström method. Advances in neural information processing systems, 32, 2019. Google Scholar
  4. Jason Altschuler, Jonathan Weed, and Philippe Rigollet. Near-linear time approximation algorithms for optimal transport via sinkhorn iteration. In Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA, pages 1964-1974, 2017. Google Scholar
  5. Sepehr Assadi, Arun Jambulapati, Yujia Jin, Aaron Sidford, and Kevin Tian. Semi-streaming bipartite matching in fewer passes and optimal space. In Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Alexandria, VA, USA, January 9 - 12, 2022, 2022. Google Scholar
  6. Aaron Bernstein, Maximilian Probst Gutenberg, and Thatchaphol Saranurak. Deterministic decremental reachability, scc, and shortest paths via directed expanders and congestion balancing. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 1123-1134, 2020. Google Scholar
  7. Digvijay Boob, Saurabh Sawlani, and Di Wang. Faster width-dependent algorithm for mixed packing and covering lps. In Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pages 15253-15262, 2019. Google Scholar
  8. Bartlomiej Bosek, Dariusz Leniowski, Piotr Sankowski, and Anna Zych. Online bipartite matching in offline time. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 384-393. IEEE, 2014. Google Scholar
  9. Yair Carmon, Yujia Jin, Aaron Sidford, and Kevin Tian. Variance reduction for matrix games. In Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pages 11377-11388, 2019. Google Scholar
  10. Yair Carmon, Yujia Jin, Aaron Sidford, and Kevin Tian. Coordinate methods for matrix games. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 283-293, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00035.
  11. Li Chen, Rasmus Kyng, Yang P Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time. arXiv preprint, 2022. URL: http://arxiv.org/abs/2203.00671.
  12. Michael B Cohen, Aleksander Madry, Dimitris Tsipras, and Adrian Vladu. Matrix scaling and balancing via box constrained newton’s method and interior point methods. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 902-913. IEEE, 2017. Google Scholar
  13. Michael B. Cohen, Aaron Sidford, and Kevin Tian. Relative lipschitzness in extragradient methods and a direct recipe for acceleration. In 12th Innovations in Theoretical Computer Science Conference, ITCS 2021, January 6-8, 2021, Virtual Conference, pages 62:1-62:18, 2021. Google Scholar
  14. Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems 26: 27th Annual Conference on Neural Information Processing Systems 2013. Proceedings of a meeting held December 5-8, 2013, Lake Tahoe, Nevada, United States, pages 2292-2300, 2013. Google Scholar
  15. David Durfee, Yu Gao, Gramoz Goranci, and Richard Peng. Fully dynamic spectral vertex sparsifiers and applications. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 914-925. ACM, 2019. Google Scholar
  16. Pavel Dvurechensky, Alexander Gasnikov, and Alexey Kroshnin. Computational optimal transport: Complexity by accelerated gradient descent is better than by sinkhorn’s algorithm. In Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholmsmässan, Stockholm, Sweden, July 10-15, 2018, pages 1366-1375, 2018. Google Scholar
  17. Yu Gao, Yang P. Liu, and Richard Peng. Fully dynamic electrical flows: Sparse maxflow faster than goldberg-rao. CoRR, abs/2101.07233, 2021. URL: http://arxiv.org/abs/2101.07233.
  18. Ankit Garg, Leonid Gurvits, Rafael Mendes de Oliveira, and Avi Wigderson. Operator scaling: Theory and applications. Found. Comput. Math., 20(2):223-290, 2020. Google Scholar
  19. Gramoz Goranci. Dynamic graph algorithms and graph sparsification: New techniques and connections. CoRR, abs/1909.06413, 2019. URL: http://arxiv.org/abs/1909.06413.
  20. Gramoz Goranci, Monika Henzinger, and Pan Peng. The power of vertex sparsifiers in dynamic graph algorithms. In Kirk Pruhs and Christian Sohler, editors, 25th Annual European Symposium on Algorithms, ESA 2017, September 4-6, 2017, Vienna, Austria, volume 87 of LIPIcs, pages 45:1-45:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  21. Gramoz Goranci, Monika Henzinger, and Pan Peng. Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs. In Yossi Azar, Hannah Bast, and Grzegorz Herman, editors, 26th Annual European Symposium on Algorithms (ESA 2018), volume 112 of Leibniz International Proceedings in Informatics (LIPIcs), pages 40:1-40:15, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Google Scholar
  22. Gramoz Goranci, Harald Räcke, Thatchaphol Saranurak, and Zihan Tan. The expander hierarchy and its applications to dynamic graph algorithms. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 2212-2228. SIAM, 2021. Google Scholar
  23. Fabrizio Grandoni, Stefano Leonardi, Piotr Sankowski, Chris Schwiegelshohn, and Shay Solomon. (1 + ε)-approximate incremental matching in constant deterministic amortized time. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1886-1898, 2019. Google Scholar
  24. Manoj Gupta. Maintaining approximate maximum matching in an incremental bipartite graph in polylogarithmic update time. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS 2014, December 15-17, 2014, New Delhi, India, pages 227-239, 2014. Google Scholar
  25. Manoj Gupta and Richard Peng. Fully dynamic (1+ ε)-approximate matchings. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 548-557, 2013. Google Scholar
  26. Maximilian Probst Gutenberg and Christian Wulff-Nilsen. Fully-dynamic all-pairs shortest paths: Improved worst-case time and space bounds. 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 2562-2574. SIAM, 2020. Google Scholar
  27. Kathrin Hanauer, Monika Henzinger, and Christian Schulz. Recent advances in fully dynamic graph algorithms. CoRR, abs/2102.11169, 2021. URL: http://arxiv.org/abs/2102.11169.
  28. 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 Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 21-30, 2015. Google Scholar
  29. Arun Jambulapati, Aaron Sidford, and Kevin Tian. A direct Õ(1/ε) iteration parallel algorithm for optimal transport. In Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pages 11355-11366, 2019. Google Scholar
  30. Jonathan A. Kelner, Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 217-226, 2014. Google Scholar
  31. Peter Kiss. Improving update times of dynamic matching algorithms from amortized to worst case. CoRR, abs/2108.10461, 2021. URL: http://arxiv.org/abs/2108.10461.
  32. Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Higher lower bounds from the 3sum conjecture. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1272-1287, 2016. Google Scholar
  33. Yin Tat Lee and Aaron Sidford. Efficient inverse maintenance and faster algorithms for linear programming. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 230-249, 2015. Google Scholar
  34. Tianyi Lin, Nhat Ho, and Michael I. Jordan. On efficient optimal transport: An analysis of greedy and accelerated mirror descent algorithms. In Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, pages 3982-3991, 2019. Google Scholar
  35. Nathan Linial, Alex Samorodnitsky, and Avi Wigderson. A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents. In Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages 644-652, 1998. Google Scholar
  36. Danupon Nanongkai, Thatchaphol Saranurak, and Christian Wulff-Nilsen. Dynamic minimum spanning forest with subpolynomial worst-case update time. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 950-961. IEEE Computer Society, 2017. Google Scholar
  37. Arkadi Nemirovski. Prox-method with rate of convergence O(1/t) for variational inequalities with lipschitz continuous monotone operators and smooth convex-concave saddle point problems. SIAM Journal on Optimization, 15(1):229-251, 2004. Google Scholar
  38. Yurii Nesterov. Smooth minimization of non-smooth functions. Math. Program., 103(1):127-152, 2005. Google Scholar
  39. Yurii Nesterov. Dual extrapolation and its applications to solving variational inequalities and related problems. Math. Program., 109(2-3):319-344, 2007. Google Scholar
  40. Jonah Sherman. Nearly maximum flows in nearly linear time. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 263-269, 2013. Google Scholar
  41. Jonah Sherman. Area-convexity, 𝓁_∞ regularization, and undirected multicommodity flow. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 452-460. ACM, 2017. Google Scholar
  42. Aaron Sidford and Kevin Tian. Coordinate methods for accelerating 𝓁_∞ regression and faster approximate maximum flow. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science, pages 922-933. IEEE, 2018. Google Scholar
  43. Jan van den Brand, Yin Tat Lee, Yang P. Liu, Thatchaphol Saranurak, Aaron Sidford, Zhao Song, and Di Wang. Minimum cost flows, mdps, and 𝓁₁-regression in nearly linear time for dense instances. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 859-869, 2021. Google Scholar
  44. François-Xavier Vialard. An elementary introduction to entropic regularization and proximal methods for numerical optimal transport, 2019. Google Scholar
  45. David Wajc. Rounding dynamic matchings against an adaptive adversary. In Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 194-207, 2020. Google Scholar
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