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Faster Approximate Linear Matroid Intersection

Author Tatsuya Terao

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LIPIcs.SWAT.2026.39.pdf
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ACM Subject Classification
  • Theory of computation → Algorithm design techniques
  • Mathematics of computing → Matroids and greedoids
Keywords
  • Linear matroid intersection
  • fast approximation algorithm

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Abstract

We consider a fast approximation algorithm for the linear matroid intersection problem. In this problem, we are given two r × n matrices M₁ and M₂, and the objective is to find a largest set of columns that are linearly independent in both M₁ and M₂. We design a (1 - ε)-approximation algorithm with time complexity Õ_{ε}(nnz(M₁) + nnz(M₂) + r_{*}^{ω}), where nnz(M_i) denotes the number of nonzero entries in M_i for i = 1, 2, r_{*} denotes the maximum size of a common independent set, and ω < 2.372 denotes the matrix multiplication exponent. Our approximation algorithm is faster than the exact algorithm by Harvey [FOCS'06 & SICOMP'09] and Cheung-Kwok-Lau [STOC'12 & JACM'13], which runs in Õ(nnz(M₁) + nnz(M₂) + n r_{*}^{ω - 1}) time.
We also develop a fast (1 - ε)-approximation algorithm for the weighted version of the linear matroid intersection problem. In fact, we design a (1 - ε)-approximation algorithm for weighted linear matroid intersection with time complexity Õ_{ε}(nnz(M₁) + nnz(M₂) + r_{*}^{ω}). Our algorithm improves upon the (1 - ε)-approximation algorithm by Huang-Kakimura-Kamiyama [SODA'16 & Math. Program.'19], which runs in Õ_{ε}(nnz(M₁) + nnz(M₂) + nr_{*}^{ω - 1}) time.
To obtain these results, we combine Quanrud’s adaptive sparsification framework [ICALP'24] with a simple yet effective method for efficiently checking whether a given vector lies in the linear span of a subset of vectors, which is of independent interest.

Cite As Get BibTex

Tatsuya Terao. Faster Approximate Linear Matroid Intersection. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 39:1-39:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SWAT.2026.39

Author Details

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" and by JSPS KAKENHI Grant Number JP24KJ1494.

Acknowledgements

We thank Yusuke Kobayashi for his insightful comments, generous support, and valuable feedback on the manuscript. In particular, his input greatly helped us establish Lemma 11. We also thank Kou Hamada for helpful conversations that contributed to the initial motivation for this work. We are grateful to Takashi Noguchi for his helpful comments, which simplified the proof of Lemma 11. Finally, we thank the anonymous reviewers for their careful reading and valuable suggestions.

References

  1. Aryan Agarwala, Yaroslav Alekseev, and Antoine Vinciguerra. Linear matroid intersection is in catalytic logspace. In Proceedings of the 17th Innovations in Theoretical Computer Science Conference (ITCS 2026), volume 362, pages 3:1-3:23, 2026. URL: https://doi.org/10.4230/LIPIcs.ITCS.2026.3.
  2. Martin Aigner and Thomas A Dowling. Matching theory for combinatorial geometries. Transactions of the American Mathematical Society, 158(1):231-245, 1971. URL: https://doi.org/10.1090/S0002-9947-1971-0286689-5.
  3. Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. More asymmetry yields faster matrix multiplication. In Proceedings of the 36th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2025), pages 2005-2039. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.63.
  4. Sepehr Assadi. A simple (1-ε)-approximation semi-streaming algorithm for maximum (weighted) matching. TheoretiCS, 4, 2025. Preliminary version in SOSA 2024. URL: https://doi.org/10.46298/theoretics.25.16.
  5. Alexander I Barvinok. New algorithms for linear k-matroid intersection and matroid k-parity problems. Mathematical Programming, 69(1):449-470, 1995. URL: https://doi.org/10.1007/BF01585571.
  6. 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.
  7. 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. URL: https://doi.org/10.1145/3564246.3585219.
  8. Joakim Blikstad and Ta-Wei Tu. Efficient matroid intersection via a batch-update auction algorithm. In Proceedings of the 8th Symposium on Simplicity in Algorithms (SOSA 2025), pages 226-237. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978315.18.
  9. 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.
  10. Cornelius Brand and Kevin Pratt. Parameterized applications of symbolic differentiation of (totally) multilinear polynomials. In Proceedings of the 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), pages 38:1-38:19, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.38.
  11. James R Bunch and John E Hopcroft. Triangular factorization and inversion by fast matrix multiplication. Mathematics of Computation, 28(125):231-236, 1974. Google Scholar
  12. 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. IEEE, 2019. URL: https://doi.org/10.1109/FOCS.2019.00072.
  13. Chandra Chekuri and Kent Quanrud. A fast approximation for maximum weight matroid intersection. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2016), pages 445-457. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch33.
  14. Ho Yee Cheung, Tsz Chiu Kwok, and Lap Chi Lau. Fast matrix rank algorithms and applications. Journal of the ACM (JACM), 60(5):1-25, 2013. Preliminary version in STOC 2012. URL: https://doi.org/10.1145/2528404.
  15. Ho Yee Cheung, Lap Chi Lau, and Kai Man Leung. Algebraic algorithms for linear matroid parity problems. ACM Transactions on Algorithms (TALG), 10(3):1-26, 2014. Preliminary version in SODA 2011. URL: https://doi.org/10.1145/2601066.
  16. William H Cunningham. Improved bounds for matroid partition and intersection algorithms. SIAM Journal on Computing, 15(4):948-957, 1986. URL: https://doi.org/10.1137/0215066.
  17. Marek Cygan, Fedor V Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  18. Aditi Dudeja and Mara Grilnberger. A weighted-to-unweighted reduction for matroid intersection. arXiv preprint arXiv:2602.15702, 2026. URL: https://doi.org/10.48550/arXiv.2602.15702.
  19. Jean-Guillaume Dumas, Clément Pernet, and Ziad Sultan. Simultaneous computation of the row and column rank profiles. In Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation (ISSAC 2013), pages 181-188, 2013. URL: https://doi.org/10.1145/2465506.2465517.
  20. Jean-Guillaume Dumas, Clément Pernet, and Ziad Sultan. Fast computation of the rank profile matrix and the generalized bruhat decomposition. Journal of Symbolic Computation, 83:187-210, 2017. URL: https://doi.org/10.1016/j.jsc.2016.11.011.
  21. Jack Edmonds. Submodular functions, matroids, and certain polyhedra. In Combinatorial Structures and Their Applications, pages 69-87, 1970. Google Scholar
  22. Jack Edmonds. Matroid intersection. In Annals of Discrete Mathematics, volume 4, pages 39-49. Elsevier, 1979. URL: https://doi.org/10.1016/S0167-5060(08)70817-3.
  23. Eduard Eiben, Tomohiro Koana, and Magnus Wahlström. Determinantal sieving. TheoretiCS, 4, 2025. Preliminary version in SODA 2024. URL: https://doi.org/10.46298/theoretics.25.21.
  24. Rūsiņš Freivalds. Fast probabilistic algorithms. In Proceedings of the 8th International Symposium on Mathematical Foundations of Computer Science (MFCS 1979), pages 57-69. Springer, 1979. Google Scholar
  25. Satoru Fujishige and Zhang Xiaodong. An efficient cost scaling algorithm for the independent assignment problem. Journal of the Operations Research Society of Japan, 38(1):124-136, 1995. Google Scholar
  26. Harold N Gabow and Matthias Stallmann. Efficient algorithms for graphic matroid intersection and parity. In Proceedings of the 12th Colloquium on Automata, Languages and Programming (ICALP 1985), pages 210-220. Springer, 1985. URL: https://doi.org/10.1007/BFb0015746.
  27. Harold N Gabow and Robert E Tarjan. Efficient algorithms for simple matroid intersection problems. In Proceedings of the 20th Annual Symposium on Foundations of Computer Science (FOCS 1979), pages 196-204. IEEE, 1979. URL: https://doi.org/10.1109/SFCS.1979.14.
  28. Harold N Gabow and Ying Xu. Efficient algorithms for independent assignment on graphic and linear matroids. In Proceedings of the 30th Annual Symposium on Foundations of Computer Science (FOCS 1989), pages 106-111. IEEE Computer Society, 1989. URL: https://doi.org/10.1109/SFCS.1989.63463.
  29. Harold N Gabow and Ying Xu. Efficient theoretic and practical algorithms for linear matroid intersection problems. Journal of Computer and System Sciences, 53(1):129-147, 1996. URL: https://doi.org/10.1006/jcss.1996.0054.
  30. Rohit Gurjar, Taihei Oki, and Roshan Raj. Fractional linear matroid matching is in Quasi-NC. In Proceedings of the 32nd Annual European Symposium on Algorithms (ESA 2024), volume 308, pages 63:1-63:14, 2024. URL: https://doi.org/10.4230/LIPIcs.ESA.2024.63.
  31. Rohit Gurjar and Thomas Thierauf. Linear matroid intersection is in Quasi-NC. computational complexity, 29(2):9, 2020. Preliminary version in STOC 2017. URL: https://doi.org/10.1007/s00037-020-00200-z.
  32. Nicholas JA Harvey. An algebraic algorithm for weighted linear matroid intersection. In Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pages 444-453, 2007. URL: http://dl.acm.org/citation.cfm?id=1283383.1283430.
  33. Nicholas JA Harvey. Algebraic algorithms for matching and matroid problems. SIAM Journal on Computing, 39(2):679-702, 2009. Preliminary version in FOCS 2006. URL: https://doi.org/10.1137/070684008.
  34. Nicholas James Alexander Harvey. Matchings, matroids and submodular functions. PhD thesis, Massachusetts Institute of Technology, 2008. Google Scholar
  35. Chien-Chung Huang, Naonori Kakimura, and Naoyuki Kamiyama. Exact and approximation algorithms for weighted matroid intersection. Mathematical Programming, 177(1-2):85-112, 2019. Preliminary version in SODA 2016. URL: https://doi.org/10.1007/s10107-018-1260-x.
  36. Claude-Pierre Jeannerod, Clément Pernet, and Arne Storjohann. Rank-profile revealing gaussian elimination and the cup matrix decomposition. Journal of Symbolic Computation, 56:46-68, 2013. URL: https://doi.org/10.1016/j.jsc.2013.04.004.
  37. Tomohiro Koana and Magnus Wahlström. Faster algorithms on linear delta-matroids. In Proceedings of 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025), volume 327, pages 62:1-62:19, 2025. URL: https://doi.org/10.4230/LIPIcs.STACS.2025.62.
  38. Eugene L Lawler. Matroid intersection algorithms. Mathematical Programming, 9(1):31-56, 1975. URL: https://doi.org/10.1007/BF01681329.
  39. Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong. A faster cutting plane method and its implications for combinatorial and convex optimization. In Proceedings of the 56th Annual Symposium on Foundations of Computer Science (FOCS 2015), pages 1049-1065. IEEE, 2015. URL: https://doi.org/10.1109/FOCS.2015.68.
  40. Dániel Marx. A parameterized view on matroid optimization problems. Theoretical Computer Science, 410(44):4471-4479, 2009. Preliminary version in ICALP 2006. URL: https://doi.org/10.1016/j.tcs.2009.07.027.
  41. Kazuki Matoya and Taihei Oki. Pfaffian pairs and parities: Counting on linear matroid intersection and parity problems. SIAM Journal on Discrete Mathematics, 36(3):2121-2158, 2022. Preliminary version in IPCO 2021. URL: https://doi.org/10.1137/21m1421751.
  42. Rajeev Motwani and Prabhakar Raghavan. Randomized Algorithms. Cambridge University Press, 1995. URL: https://doi.org/10.1017/cbo9780511814075.
  43. Harihar Narayanan, Huzur Saran, and Vijay V Vazirani. Randomized parallel algorithms for matroid union and intersection, with applications to arborescences and edge-disjoint spanning trees. SIAM Journal on Computing, 23(2):387-397, 1994. Preliminary version in SODA 1992. URL: https://doi.org/10.1137/S0097539791195245.
  44. Huy L Nguyen. Fast greedy for linear matroids. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019), pages 516-524. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.32.
  45. Huy L Nguyen. A note on Cunningham’s algorithm for matroid intersection. arXiv preprint arXiv:1904.04129, 2019. URL: https://doi.org/10.48550/arXiv.1904.04129.
  46. Taihei Oki and Tasuku Soma. Algebraic algorithms for fractional linear matroid parity via noncommutative rank. SIAM Journal on Computing, 54(1):134-162, 2025. Preliminary version in SODA 2023. URL: https://doi.org/10.1137/22m1537096.
  47. James G Oxley. Matroid theory, volume 3. Oxford University Press, USA, 2006. Google Scholar
  48. Kent Quanrud. Adaptive sparsification for matroid intersection. In Proceedings of the 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024), volume 297, pages 118:1-118:20, 2024. URL: https://doi.org/10.4230/LIPIcs.ICALP.2024.118.
  49. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer, 2003. Google Scholar
  50. Jacob T Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM (JACM), 27(4):701-717, 1980. URL: https://doi.org/10.1145/322217.322225.
  51. Maiko Shigeno and Satoru Iwata. A dual approximation approach to weighted matroid intersection. Operations Research Letters, 18(3):153-156, 1995. URL: https://doi.org/10.1016/0167-6377(95)00047-X.
  52. Arne Storjohann and Shiyun Yang. Linear independence oracles and applications to rectangular and low rank linear systems. In Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (ISSAC 2014), pages 381-388, 2014. URL: https://doi.org/10.1145/2608628.2608673.
  53. Tatsuya Terao. Deterministic (2/3 - ε)-approximation of matroid intersection using nearly-linear independence-oracle queries. In Proceedings of the 19th International Symposium on Algorithms and Data Structures (WADS 2025), volume 349, pages 50:1-50:18, 2025. URL: https://doi.org/10.4230/LIPIcs.WADS.2025.50.
  54. Tatsuya Terao. Faster approximate linear matroid intersection. arXiv preprint arXiv:2604.11725, 2026. URL: https://doi.org/10.48550/arXiv.2604.11725.
  55. Nobuaki Tomizawa and Masao Iri. An algorithm for determining the rank of a triple matrix product AXB with application to the problem of discerning the existence of the unique solution in a network. Electronics and Communications in Japan (Scripta Electronica Japonica II), 57(11):50-57, 1974. Google Scholar
  56. 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.
  57. Jan van den Brand. Unifying matrix data structures: Simplifying and speeding up iterative algorithms. In Proceedings of Symposium on Simplicity in Algorithms (SOSA 2021), pages 1-13. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976496.1.
  58. Jan van den Brand, Danupon Nanongkai, and Thatchaphol Saranurak. Dynamic matrix inverse: Improved algorithms and matching conditional lower bounds. In Proceedings of the 60th Annual Symposium on Foundations of Computer Science (FOCS 2019), pages 456-480. IEEE, 2019. URL: https://doi.org/10.1109/FOCS.2019.00036.
  59. Ying Xu and Harold N Gabow. Fast algorithms for transversal matroid intersection problems. In Proceedings of the 5th International Symposium on Algorithms and Computation (ISAAC 1994), pages 625-633. Springer, 1994. URL: https://doi.org/10.1007/3-540-58325-4_231.
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