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Subquadratic Submodular Maximization with a General Matroid Constraint

Authors Yusuke Kobayashi , Tatsuya Terao

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Author Details

Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Tatsuya Terao
  • Research Institute for Mathematical Sciences, Kyoto University, Japan

Funding

This work was partially supported by the joint project of Kyoto University and Toyota Motor Corporation, titled "Advanced Mathematical Science for Mobility Society", by JST ERATO Grant Number JPMJER2310, and by JSPS KAKENHI Grant Numbers JP20K11692, JP22H05001, JP24KJ1494, and JP24K02901.

Acknowledgements

The authors thank the three anonymous reviewers for their valuable comments.

Cite As Get BibTex

Yusuke Kobayashi and Tatsuya Terao. Subquadratic Submodular Maximization with a General Matroid Constraint. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 100:1-100:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.100

Abstract

We consider fast algorithms for monotone submodular maximization with a general matroid constraint. We present a randomized (1 - 1/e - ε)-approximation algorithm that requires Õ_{ε}(√r n) independence oracle and value oracle queries, where n is the number of elements in the matroid and r ≤ n is the rank of the matroid. This improves upon the previously best algorithm by Buchbinder-Feldman-Schwartz [Mathematics of Operations Research 2017] that requires Õ_{ε}(r² + √rn) queries.
Our algorithm is based on continuous relaxation, as with other submodular maximization algorithms in the literature. To achieve subquadratic query complexity, we develop a new rounding algorithm, which is our main technical contribution. The rounding algorithm takes as input a point represented as a convex combination of t bases of a matroid and rounds it to an integral solution. Our rounding algorithm requires Õ(r^{3/2} t) independence oracle queries, while the previously best rounding algorithm by Chekuri-Vondrák-Zenklusen [FOCS 2010] requires O(r² t) independence oracle queries. A key idea in our rounding algorithm is to use a directed cycle of arbitrary length in an auxiliary graph, while the algorithm of Chekuri-Vondrák-Zenklusen focused on directed cycles of length two.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithm design techniques
  • Theory of computation → Submodular optimization and polymatroids
Keywords
  • submodular maximization
  • matroid constraint
  • approximation algorithm
  • rounding algorithm
  • query complexity

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References

  1. Alexander A Ageev and Maxim I Sviridenko. Pipage rounding: A new method of constructing algorithms with proven performance guarantee. Journal of Combinatorial Optimization, 8:307-328, 2004. URL: https://doi.org/10.1023/B:JOCO.0000038913.96607.c2.
  2. Yossi Azar and Iftah Gamzu. Efficient submodular function maximization under linear packing constraints. In Proceedings of the 39th International Colloquium on Automata, Languages, and Programming (ICALP 2012), pages 38-50, 2012. URL: https://doi.org/10.1007/978-3-642-31594-7_4.
  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 2014), pages 1497-1514, 2014. URL: https://doi.org/10.1137/1.9781611973402.110.
  4. Eric Balkanski, Aviad Rubinstein, and Yaron Singer. An optimal approximation for submodular maximization under a matroid constraint in the adaptive complexity model. Operations Research, 70(5):2967-2981, 2022. URL: https://doi.org/10.1287/opre.2021.2170.
  5. Eric Balkanski and Yaron Singer. The adaptive complexity of maximizing a submodular function. In Proceedings of the 50th annual ACM SIGACT Symposium on Theory of Computing (STOC 2018), pages 1138-1151, 2018. URL: https://doi.org/10.1145/3188745.3188752.
  6. Kiarash Banihashem, Leyla Biabani, Samira Goudarzi, MohammadTaghi Hajiaghayi, Peyman Jabbarzade, and Morteza Monemizadeh. Dynamic algorithms for matroid submodular maximization. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2024), pages 3485-3533, 2024. URL: https://doi.org/10.1137/1.9781611977912.125.
  7. Rafael da Ponte Barbosa, Alina Ene, Huy L Nguy~ên, and Justin Ward. A new framework for distributed submodular maximization. In Proceedings of the 57th Annual Symposium on Foundations of Computer Science (FOCS 2016), pages 645-654, 2016. URL: https://doi.org/10.1109/FOCS.2016.74.
  8. Joakim Blikstad. Breaking O(nr) for matroid intersection. In Proceedings of the 48th International Colloquium on Automata, Languages, and Programming (ICALP 2022), volume 198, pages 31:1-31:17, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.31.
  9. Joakim Blikstad, Sagnik Mukhopadhyay, Danupon Nanongkai, and Ta-Wei Tu. Fast algorithms via dynamic-oracle matroids. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing (STOC 2023), pages 1229-1242, 2023. arXiv version is arXiv:2302.09796. URL: https://doi.org/10.1145/3564246.3585219.
  10. Joakim Blikstad, Jan van den Brand, Sagnik Mukhopadhyay, and Danupon Nanongkai. Breaking the quadratic barrier for matroid intersection. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2021), pages 421-432, 2021. URL: https://doi.org/10.1145/3406325.3451092.
  11. Niv Buchbinder, Moran Feldman, Joseph Naor, and Roy Schwartz. Submodular maximization with cardinality constraints. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2014), pages 1433-1452, 2014. URL: https://doi.org/10.1137/1.9781611973402.106.
  12. Niv Buchbinder, Moran Feldman, and Roy Schwartz. Comparing apples and oranges: Query trade-off in submodular maximization. Mathematics of Operations Research, 42(2):308-329, 2017. URL: https://doi.org/10.1287/moor.2016.0809.
  13. Gruia Calinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing, 40(6):1740-1766, 2011. URL: https://doi.org/10.1137/080733991.
  14. Deeparnab Chakrabarty, Yin Tat Lee, Aaron Sidford, Sahil Singla, and Sam Chiu-wai Wong. Faster matroid intersection. In Proceedings of the 60th Annual Symposium on Foundations of Computer Science (FOCS 2019), pages 1146-1168, 2019. URL: https://doi.org/10.1109/FOCS.2019.00072.
  15. Chandra Chekuri, TS Jayram, and Jan Vondrák. On multiplicative weight updates for concave and submodular function maximization. In Proceedings of the 6th Conference on Innovations in Theoretical Computer Science (ITCS 2015), pages 201-210, 2015. URL: https://doi.org/10.1145/2688073.2688086.
  16. Chandra Chekuri and Kent Quanrud. Parallelizing greedy for submodular set function maximization in matroids and beyond. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC 2019), pages 78-89, 2019. URL: https://doi.org/10.1145/3313276.3316406.
  17. Chandra Chekuri, Kent Quanrud, and Manuel R Torres. Fast approximation algorithms for bounded degree and crossing spanning tree problems. In Proceedings of the 24th International Conference on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2021), volume 207, pages 24:1-24:21, 2021. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.24.
  18. Chandra Chekuri, Jan Vondrák, and Rico Zenklusen. Dependent randomized rounding via exchange properties of combinatorial structures. In Proceedings of the 51st Annual Symposium on Foundations of Computer Science (FOCS 2010), pages 575-584, 2010. URL: https://doi.org/10.1109/FOCS.2010.60.
  19. Xi Chen and Binghui Peng. On the complexity of dynamic submodular maximization. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (FOCS 2022), pages 1685-1698, 2022. URL: https://doi.org/10.1145/3519935.3519951.
  20. Alina Ene and Huy L. Nguy~ên. A nearly-linear time algorithm for submodular maximization with a knapsack constraint. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), volume 132, pages 53:1-53:12, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.53.
  21. Alina Ene and Huy L Nguy~ên. Submodular maximization with nearly-optimal approximation and adaptivity in nearly-linear time. In Proceedings of the 30-th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019), pages 274-282, 2019. URL: https://doi.org/10.1137/1.9781611975482.18.
  22. Alina Ene and Huy L. Nguy~ên. Towards nearly-linear time algorithms for submodular maximization with a matroid constraint. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), volume 132, pages 54:1-54:14, 2019. arXiv version is arXiv:1811.07464. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.54.
  23. Alina Ene, Huy L Nguy~ên, and Adrian Vladu. Submodular maximization with matroid and packing constraints in parallel. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC 2019), pages 90-101, 2019. URL: https://doi.org/10.1145/3313276.3316389.
  24. Uriel Feige. A threshold of ln n for approximating set cover. Journal of the ACM (JACM), 45(4):634-652, 1998. URL: https://doi.org/10.1145/285055.285059.
  25. Yuval Filmus and Justin Ward. A tight combinatorial algorithm for submodular maximization subject to a matroid constraint. In Proceedings of the 53rd Annual Symposium on Foundations of Computer Science (FOCS 2012), pages 659-668, 2012. URL: https://doi.org/10.1109/FOCS.2012.55.
  26. ML Fisher, GL Nemhauser, and LA Wolsey. An analysis of approximations for maximizing submodular set functions-II. Mathematical Programming Studies, 8:73-87, 1978. URL: https://doi.org/10.1007/BFb0121195.
  27. Monika Henzinger, Paul Liu, Jan Vondrák, and Da Wei Zheng. Faster submodular maximization for several classes of matroids. In Proceedings of the 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), volume 261, pages 74:1-74:18, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.74.
  28. Felix Hommelsheim, Nicole Megow, Komal Muluk, and Britta Peis. Recoverable robust optimization with commitment, 2023. URL: https://arxiv.org/abs/2306.08546.
  29. Silvio Lattanzi, Slobodan Mitrović, Ashkan Norouzi-Fard, Jakub M Tarnawski, and Morteza Zadimoghaddam. Fully dynamic algorithm for constrained submodular optimization. Advances in Neural Information Processing Systems 33: Proceedings of the 34th Annual Conference on Neural Information Processing Systems (Neurips 2020), 33:12923-12933, 2020. Google Scholar
  30. Wenxin Li, Moran Feldman, Ehsan Kazemi, and Amin Karbasi. Submodular maximization in clean linear time. Advances in Neural Information Processing Systems 35: Proceedings of the 36th Annual Conference on Neural Information Processing Systems (Neurips 2022), 35:17473-17487, 2022. Google Scholar
  31. Paul Liu and Jan Vondrák. Submodular optimization in the MapReduce model. In Proceedings of the 2nd Symposium on Simplicity in Algorithms (SOSA 2019), volume 69, pages 18:1-18:10, 2019. URL: https://doi.org/10.4230/OASIcs.SOSA.2019.18.
  32. Baharan Mirzasoleiman, Ashwinkumar Badanidiyuru, Amin Karbasi, Jan Vondrák, and Andreas Krause. Lazier than lazy greedy. In Proceedings of the 29th AAAI Conference on Artificial Intelligence (AAAI 2015), volume 29(1), 2015. URL: https://doi.org/10.1609/aaai.v29i1.9486.
  33. Morteza Monemizadeh. Dynamic submodular maximization. Advances in Neural Information Processing Systems 33: Proceedings of the 34th Annual Conference on Neural Information Processing Systems (Neurips 2020), 33:9806-9817, 2020. Google Scholar
  34. George L Nemhauser and Laurence A Wolsey. Best algorithms for approximating the maximum of a submodular set function. Mathematics of operations research, 3(3):177-188, 1978. URL: https://doi.org/10.1287/moor.3.3.177.
  35. George L Nemhauser, Laurence A Wolsey, and Marshall L Fisher. An analysis of approximations for maximizing submodular set functions-I. Mathematical Programming, 14:265-294, 1978. URL: https://doi.org/10.1007/BF01588971.
  36. Huy L Nguy~ên. A note on Cunningham’s algorithm for matroid intersection. arXiv preprint arXiv:1904.04129, 2019. URL: https://doi.org/10.48550/arXiv.1904.04129.
  37. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer, 2003. Google Scholar
  38. Tatsuya Terao. Faster matroid partition algorithms. In Proceedings of the 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), volume 261, pages 104:1-104:20, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.104.
  39. Ta-Wei Tu. Subquadratic weighted matroid intersection under rank oracles. In Proceedings of the 33rd International Symposium on Algorithms and Computation (ISAAC 2022), volume 248, pages 63:1-63:14, 2022. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2022.63.
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