On Optimization and Counting of Non-Broken Bases of Matroids

Authors Dorna Abdolazimi, Kasper Lindberg, Shayan Oveis Gharan

Thumbnail PDF


  • Filesize: 0.71 MB
  • 14 pages

Document Identifiers

Author Details

Dorna Abdolazimi
  • University of Washington, Seattle, WA, USA
Kasper Lindberg
  • University of Washington, Seattle, WA, USA
Shayan Oveis Gharan
  • University of Washington, Seattle, WA, USA

Cite AsGet BibTex

Dorna Abdolazimi, Kasper Lindberg, and Shayan Oveis Gharan. On Optimization and Counting of Non-Broken Bases of Matroids. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Given a matroid M = (E,I), and a total ordering over the elements E, a broken circuit is a circuit where the smallest element is removed and an NBC independent set is an independent set in I with no broken circuit. The set of NBC independent sets of any matroid M define a simplicial complex called the broken circuit complex which has been the subject of intense study in combinatorics. Recently, Adiprasito, Huh and Katz showed that the face of numbers of any broken circuit complex form a log-concave sequence, proving a long-standing conjecture of Rota. We study counting and optimization problems on NBC bases of a generic matroid. We find several fundamental differences with the independent set complex: for example, we show that it is NP-hard to find the max-weight NBC base of a matroid or that the convex hull of NBC bases of a matroid has edges of arbitrary large length. We also give evidence that the natural down-up walk on the space of NBC bases of a matroid may not mix rapidly by showing that for some family of matroids it is NP-hard to count the number of NBC bases after certain conditionings.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Theory of computation → Computational complexity and cryptography
  • Complexity
  • Hardness
  • Optimization
  • Counting
  • Random walk
  • Local to Global
  • Matroids


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


  1. Karim Adiprasito, June Huh, and Eric Katz. Hodge theory for combinatorial geometries, 2018. URL: https://arxiv.org/abs/1511.02888.
  2. Vedat Levi Alev and Lap Chi Lau. Improved analysis of higher order random walks and applications. In Proceedings of the 52nd Annual ACM Symposium on Theory of Computing (STOC), 2020. Google Scholar
  3. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials ii: high-dimensional walks and an fpras for counting bases of a matroid. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 1-12, 2019. Google Scholar
  4. Brian Benson, Deeparnab Chakrabarty, and Prasad Tetali. g-parking functions, acyclic orientations and spanning trees, 2010. URL: https://arxiv.org/abs/0801.1114.
  5. Tom Brylawski. The broken-circuit complex. Trans. Amer. Math. Soc., 234(2):417-433, 1977. Google Scholar
  6. Charles J Colbourn and William R Pulleyblank. Matroid steiner problems, the tutte polynomial and network reliability. Journal of Combinatorial Theory, Series B, 47(1):20-31, 1989. URL: https://doi.org/10.1016/0095-8956(89)90062-2.
  7. Mary Cryan, Heng Guo, and Giorgos Mousa. Modified log-sobolev inequalities for strongly log-concave distributions, 2020. URL: https://arxiv.org/abs/1903.06081.
  8. Ewan Davies and Will Perkins. Approximately counting independent sets of a given size in bounded-degree graphs. arXiv preprint, 2021. URL: https://arxiv.org/abs/2102.04984.
  9. Yotam Dikstein, Irit Dinur, Yuval Filmus, and Prahladh Harsha. Boolean Function Analysis on High-Dimensional Expanders. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018), volume 116 of Leibniz International Proceedings in Informatics (LIPIcs), pages 38:1-38:20, 2018. Google Scholar
  10. I. Dinur and T. Kaufman. High dimensional expanders imply agreement expanders. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 974-985, 2017. Google Scholar
  11. Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. Inapproximability of the partition function for the antiferromagnetic ising and hard-core models. Combinatorics, Probability and Computing, 25(4):500-559, 2016. Google Scholar
  12. I.M Gelfand, R.M Goresky, R.D MacPherson, and V.V Serganova. Combinatorial geometries, convex polyhedra, and schubert cells. Advances in Mathematics, 63(3):301-316, 1987. URL: https://doi.org/10.1016/0001-8708(87)90059-4.
  13. Emeric Gioan and Michel Las Vergnas. The active bijection for graphs. Advances in Applied Mathematics, 104:165-236, 2019. URL: https://doi.org/10.1016/j.aam.2018.11.001.
  14. Roy Gotlib and Tali Kaufman. No where to go but high: A perspective on high dimensional expanders, 2023. URL: https://arxiv.org/abs/2304.10106.
  15. Mark Jerrum and Alistair Sinclair. Approximating the permanent. SIAM Journal on Computing, 18(6):1149-1178, 1989. URL: https://doi.org/10.1137/0218077.
  16. Richard M. Karp. Reducibility among Combinatorial Problems, pages 85-103. Springer US, Boston, MA, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  17. Tali Kaufman and Izhar Oppenheim. High order random walks: Beyond spectral gap. In APPROX/RANDOM, pages 47:1-47:17, 2018. Google Scholar
  18. Milena Mihail and Umesh Vazirani. On the expansion of 0-1 polytopes. preprint, 3461, 1989. Google Scholar
  19. Geoffrey Ramseyer, Ashish Goel, and David Mazières. Liquidity in credit networks with constrained agents. In Proceedings of The Web Conference 2020. ACM, April 2020. URL: https://doi.org/10.1145/3366423.3380276.
  20. Allan Sly. Computational transition at the uniqueness threshold. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pages 287-296. IEEE, 2010. Google Scholar
  21. Allan Sly and Nike Sun. Counting in two-spin models on d-regular graphs. Annals of Probability, 42(6):2383-2416, 2014. Google Scholar
  22. Richard P. Stanley. Acyclic orientations of graphs. Discrete Mathematics, 5(2):171-178, 1973. URL: https://doi.org/10.1016/0012-365X(73)90108-8.
  23. Richard P Stanley et al. An introduction to hyperplane arrangements. Geometric combinatorics, 13(389-496):24, 2004. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail