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Faster Submodular Maximization for Several Classes of Matroids

Authors Monika Henzinger, Paul Liu , Jan Vondrák , Da Wei Zheng

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Author Details

Monika Henzinger
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Paul Liu
  • Stanford University, CA, USA
Jan Vondrák
  • Stanford University, CA, USA
Da Wei Zheng
  • University of Illinois Urbana-Champaign, IL, USA

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Monika Henzinger, Paul Liu, Jan Vondrák, and Da Wei Zheng. Faster Submodular Maximization for Several Classes of Matroids. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 74:1-74:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


The maximization of submodular functions have found widespread application in areas such as machine learning, combinatorial optimization, and economics, where practitioners often wish to enforce various constraints; the matroid constraint has been investigated extensively due to its algorithmic properties and expressive power. Though tight approximation algorithms for general matroid constraints exist in theory, the running times of such algorithms typically scale quadratically, and are not practical for truly large scale settings. Recent progress has focused on fast algorithms for important classes of matroids given in explicit form. Currently, nearly-linear time algorithms only exist for graphic and partition matroids [Alina Ene and Huy L. Nguyen, 2019]. In this work, we develop algorithms for monotone submodular maximization constrained by graphic, transversal matroids, or laminar matroids in time near-linear in the size of their representation. Our algorithms achieve an optimal approximation of 1-1/e-ε and both generalize and accelerate the results of Ene and Nguyen [Alina Ene and Huy L. Nguyen, 2019]. In fact, the running time of our algorithm cannot be improved within the fast continuous greedy framework of Badanidiyuru and Vondrák [Ashwinkumar Badanidiyuru and Jan Vondrák, 2014]. To achieve near-linear running time, we make use of dynamic data structures that maintain bases with approximate maximum cardinality and weight under certain element updates. These data structures need to support a weight decrease operation and a novel Freeze operation that allows the algorithm to freeze elements (i.e. force to be contained) in its basis regardless of future data structure operations. For the laminar matroid, we present a new dynamic data structure using the top tree interface of Alstrup, Holm, de Lichtenberg, and Thorup [Stephen Alstrup et al., 2005] that maintains the maximum weight basis under insertions and deletions of elements in O(log n) time. This data structure needs to support certain subtree query and path update operations that are performed every insertion and deletion that are non-trivial to handle in conjunction. For the transversal matroid the Freeze operation corresponds to requiring the data structure to keep a certain set S of vertices matched, a property that we call S-stability. While there is a large body of work on dynamic matching algorithms, none are S-stable and maintain an approximate maximum weight matching under vertex updates. We give the first such algorithm for bipartite graphs with total running time linear (up to log factors) in the number of edges.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Mathematics of computing → Submodular optimization and polymatroids
  • Theory of computation → Dynamic graph algorithms
  • submodular optimization
  • dynamic data structures
  • matching algorithms


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  1. Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 434-443. IEEE Computer Society, 2014. URL:
  2. Stephen Alstrup, Jacob Holm, Kristian de Lichtenberg, and Mikkel Thorup. Maintaining information in fully dynamic trees with top trees. ACM Trans. Algorithms, 1(2):243-264, 2005. URL:
  3. Moshe Babaioff, Jason Hartline, and Robert Kleinberg. Selling banner ads: Online algorithms with buyback. In Fourth workshop on ad auctions, 2008. Google Scholar
  4. Moshe Babaioff, Nicole Immorlica, and Robert Kleinberg. Matroids, secretary problems, and online mechanisms. In Nikhil Bansal, Kirk Pruhs, and Clifford Stein, editors, Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, New Orleans, Louisiana, USA, January 7-9, 2007, pages 434-443. SIAM, 2007. URL:
  5. Ashwinkumar Badanidiyuru, Baharan Mirzasoleiman, Amin Karbasi, and Andreas Krause. Streaming submodular maximization: massive data summarization on the fly. In Sofus A. Macskassy, Claudia Perlich, Jure Leskovec, Wei Wang, and Rayid Ghani, editors, The 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '14, New York, NY, USA - August 24 - 27, 2014, pages 671-680. ACM, 2014. URL:
  6. Ashwinkumar Badanidiyuru and Jan Vondrák. Fast algorithms for maximizing submodular functions. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1497-1514. SIAM, 2014. URL:
  7. Jeff A. Bilmes. Submodularity in machine learning and artificial intelligence. CoRR, abs/2202.00132, 2022. URL:
  8. Bartlomiej Bosek, Dariusz Leniowski, Piotr Sankowski, and Anna Zych. Online bipartite matching in offline time. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 384-393. IEEE Computer Society, 2014. URL:
  9. Niv Buchbinder, Moran Feldman, Joseph Naor, and Roy Schwartz. Submodular maximization with cardinality constraints. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1433-1452. SIAM, 2014. URL:
  10. Niv Buchbinder, Moran Feldman, and Roy Schwartz. Comparing apples and oranges: Query trade-off in submodular maximization. Math. Oper. Res., 42(2):308-329, 2017. URL:
  11. 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:
  12. Alina Ene and Huy L. Nguyen. Towards nearly-linear time algorithms for submodular maximization with a matroid constraint. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 54:1-54:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL:
  13. Uriel Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634-652, 1998. URL:
  14. Michael L. Fredman and Robert Endre Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM, 34(3):596-615, 1987. URL:
  15. 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. IEEE Computer Society, 2013. URL:
  16. Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Rocco A. Servedio and Ronitt Rubinfeld, editors, 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. ACM, 2015. URL:
  17. Monika Henzinger, Ami Paz, and Stefan Schmid. On the complexity of weight-dynamic network algorithms. In Zheng Yan, Gareth Tyson, and Dimitrios Koutsonikolas, editors, IFIP Networking Conference, IFIP Networking 2021, Espoo and Helsinki, Finland, June 21-24, 2021, pages 1-9. IEEE, 2021. URL:
  18. Ehsan Kazemi, Marko Mitrovic, Morteza Zadimoghaddam, Silvio Lattanzi, and Amin Karbasi. Submodular streaming in all its glory: Tight approximation, minimum memory and low adaptive complexity. In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine Learning Research, pages 3311-3320. PMLR, 2019. URL:
  19. Hung Le, Lazar Milenkovic, Shay Solomon, and Virginia Vassilevska Williams. Dynamic matching algorithms under vertex updates. In Mark Braverman, editor, 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 31 - February 3, 2022, Berkeley, CA, USA, volume 215 of LIPIcs, pages 96:1-96:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL:
  20. Wenxin Li, Moran Feldman, Ehsan Kazemi, and Amin Karbasi. Submodular maximization in clean linear time. CoRR, abs/2006.09327, 2022. URL:
  21. Paul Liu and Jan Vondrák. Submodular optimization in the mapreduce model. 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 18:1-18:10. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL:
  22. 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
  23. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer, 2003. Google Scholar
  24. Robert Endre Tarjan. Efficiency of a good but not linear set union algorithm. J. ACM, 22(2):215-225, 1975. URL:
  25. Mark Wilhelm, Ajith Ramanathan, Alexander Bonomo, Sagar Jain, Ed H. Chi, and Jennifer Gillenwater. Practical diversified recommendations on youtube with determinantal point processes. In Alfredo Cuzzocrea, James Allan, Norman W. Paton, Divesh Srivastava, Rakesh Agrawal, Andrei Z. Broder, Mohammed J. Zaki, K. Selçuk Candan, Alexandros Labrinidis, Assaf Schuster, and Haixun Wang, editors, Proceedings of the 27th ACM International Conference on Information and Knowledge Management, CIKM 2018, Torino, Italy, October 22-26, 2018, pages 2165-2173. ACM, 2018. URL:
  26. Zhou Xu and Brian Rodrigues. A 3/2-approximation algorithm for the multiple tsp with a fixed number of depots. INFORMS Journal on Computing, 27(4):636-645, 2015. Google Scholar
  27. Da Wei Zheng and Monika Henzinger. Multiplicative auction algorithm for approximate maximum weight bipartite matching. CoRR, abs/2301.09217, 2023. URL:
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