Determinants vs. Algebraic Branching Programs

Authors Abhranil Chatterjee, Mrinal Kumar, Ben Lee Volk

Thumbnail PDF


  • Filesize: 0.63 MB
  • 13 pages

Document Identifiers

Author Details

Abhranil Chatterjee
  • Indian Statistical Institute, Kolkata, India
Mrinal Kumar
  • Tata Institute of Fundamental Research, Mumbai, India
Ben Lee Volk
  • Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel

Cite AsGet BibTex

Abhranil Chatterjee, Mrinal Kumar, and Ben Lee Volk. Determinants vs. Algebraic Branching Programs. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We show that for every homogeneous polynomial of degree d, if it has determinantal complexity at most s, then it can be computed by a homogeneous algebraic branching program (ABP) of size at most O(d⁵s). Moreover, we show that for most homogeneous polynomials, the width of the resulting homogeneous ABP is just s-1 and the size is at most O(ds). Thus, for constant degree homogeneous polynomials, their determinantal complexity and ABP complexity are within a constant factor of each other and hence, a super-linear lower bound for ABPs for any constant degree polynomial implies a super-linear lower bound on determinantal complexity; this relates two open problems of great interest in algebraic complexity. As of now, super-linear lower bounds for ABPs are known only for polynomials of growing degree [Mrinal Kumar, 2019; Prerona Chatterjee et al., 2022], and for determinantal complexity the best lower bounds are larger than the number of variables only by a constant factor [Mrinal Kumar and Ben Lee Volk, 2022]. While determinantal complexity and ABP complexity are classically known to be polynomially equivalent [Meena Mahajan and V. Vinay, 1997], the standard transformation from the former to the latter incurs a polynomial blow up in size in the process, and thus, it was unclear if a super-linear lower bound for ABPs implies a super-linear lower bound on determinantal complexity. In particular, a size preserving transformation from determinantal complexity to ABPs does not appear to have been known prior to this work, even for constant degree polynomials.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Circuit complexity
  • Determinant
  • Algebraic Branching Program
  • Lower Bounds
  • Singular Variety


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


  1. Jarod Alper, Tristram Bogart, and Mauricio Velasco. A lower bound for the determinantal complexity of a hypersurface. Found. Comput. Math., 17(3):829-836, 2017. URL:
  2. Walter Baur and Volker Strassen. The complexity of partial derivatives. Theoretical Computer Science, 22:317-330, 1983. URL:
  3. Stuart J. Berkowitz. On computing the determinant in small parallel time using a small number of processors. Information Processing Letters, 18(3):147-150, 1984. URL:
  4. Jin-Yi Cai, Xi Chen, and Dong Li. Quadratic lower bound for permanent vs. determinant in any characteristic. Comput. Complex., 19(1):37-56, 2010. URL:
  5. Abhranil Chatterjee, Mrinal Kumar, and Ben Lee Volk. Determinants vs. algebraic branching programs. CoRR, abs/2308.04599, 2023. URL:
  6. Prerona Chatterjee, Mrinal Kumar, Adrian She, and Ben Lee Volk. Quadratic lower bounds for algebraic branching programs and formulas. Comput. Complex., 31(2):8, 2022. URL:
  7. Kyriakos Kalorkoti. A Lower Bound for the Formula Size of Rational Functions. SIAM Journal of Computing, 14(3):678-687, 1985. URL:
  8. Mauricio Karchmer and Avi Wigderson. On span programs. In Proceedings of the 8th Annual Structure in Complexity Theory Conference (Structures 1993), pages 102-111. IEEE Computer Society, 1993. URL:
  9. Mrinal Kumar. A quadratic lower bound for homogeneous algebraic branching programs. Computational Complexity, 28(3):409-435, 2019. URL:
  10. Mrinal Kumar and Ben Lee Volk. A lower bound on determinantal complexity. Comput. Complex., 31(2):12, 2022. URL:
  11. J.M. Landsberg and Nicolas Ressayre. Permanent v. determinant: An exponential lower bound assuming symmetry and a potential path towards valiant’s conjecture. Differential Geometry and its Applications, 55:146-166, 2017. URL:
  12. Satyanarayana V. Lokam. Complexity lower bounds using linear algebra. Found. Trends Theor. Comput. Sci., 4(1-2):1-155, 2009. URL:
  13. Meena Mahajan and V. Vinay. A combinatorial algorithm for the determinant. In Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1997), pages 730-738, 1997. Available on URL:
  14. Thierry Mignon and Nicolas Ressayre. A quadratic bound for the determinant and permanent problem. International Mathematics Research Notes, 2004(79):4241-4253, 2004. Available on URL:
  15. Eduard Ivanovich Nechiporuk. On a boolean function. Dokl. Akad. Nauk SSSR, 169:765-766, 1966. URL:
  16. Ran Raz. Elusive functions and lower bounds for arithmetic circuits. Theory of Computing, 6(7):135-177, 2010. URL:
  17. Ramprasad Saptharishi. A survey of lower bounds in arithmetic circuit complexity. Github survey, 2015. URL:
  18. Victor Shoup and Roman Smolensky. Lower bounds for polynomial evaluation and interpolation problems. Comput. Complex., 6(4):301-311, 1997. URL:
  19. Volker Strassen. Die berechnungskomplexität von elementarsymmetrischen funktionen und von interpolationskoeffizienten. Numerische Mathematik, 20(3):238-251, June 1973. URL:
  20. Leslie G. Valiant. On non-linear lower bounds in computational complexity. In William C. Rounds, Nancy Martin, Jack W. Carlyle, and Michael A. Harrison, editors, Proceedings of the 7th Annual ACM Symposium on Theory of Computing, May 5-7, 1975, Albuquerque, New Mexico, USA, pages 45-53. ACM, 1975. URL:
  21. Leslie G. Valiant. Graph-theoretic arguments in low-level complexity. In Jozef Gruska, editor, Proceedings of the 2nd International Symposium on the Mathematical Foundations of Computer Science (MFCS 1977), volume 53 of Lecture Notes in Computer Science, pages 162-176. Springer, 1977. URL:
  22. Leslie G. Valiant. Completeness Classes in Algebra. In Proceedings of the 11th Annual ACM Symposium on Theory of Computing (STOC 1979), pages 249-261, 1979. URL:
  23. Joachim von zur Gathen. Permanent and determinant. Linear Algebra and its Applications, 96:87-100, 1987. URL:
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