Document Open Access Logo

On the Tension Between Full-Rankness and Self-Reducibility for Set-Multilinear Polynomials

Author Deepanshu Kush

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2026.69.pdf
  • Filesize: 0.83 MB
  • 17 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Circuit complexity
Keywords
  • Algebraic formula lower bounds
  • set-multilinear formulas
  • iterated matrix multiplication
  • rank methods
  • hardness escalation
  • self-reducibility
  • barriers

Metrics

Abstract

In this paper, we rule out a natural programme for proving VF ≠ VNP via set-multilinear formula lower bounds. The programme combines two ingredients present in the literature: the n^Ω(log n) set-multilinear formula lower bound of Kush and Saraf (CCC 2022) for any full-rank polynomial, and an IMM-style self-reducibility that propagates such a bound to small degree, where Raz’s set-multilinearisation (J. ACM 2013) converts it to a general formula lower bound. Each ingredient has been realised separately, yet no polynomial family is known to combine them. We prove that no such family can exist: under any polynomial-width IMM-style self-reducibility - i.e., a small-width expression g = ∑_{k=1}^w L_k ⋅ R_k with each summand factoring across a balanced split - full-rankness forces width n^Ω(d), and even approximate full-rankness across a near-balanced split forces width n^Ω(√d). This rules out the full-rank/self-reducible route to VF ≠ VNP.

Cite As Get BibTex

Deepanshu Kush. On the Tension Between Full-Rankness and Self-Reducibility for Set-Multilinear Polynomials. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 69:1-69:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.69

Author Details

Deepanshu Kush
  • Department of Computer Science and Technology, University of Cambridge, UK
  • Sidney Sussex College, Cambridge, UK

Funding

This research is supported by Tom Gur’s ERC Starting Grant 101163189 and UKRI Future Leaders Fellowship MR/X023583/1.

References

  1. Richard P. Brent. The parallel evaluation of general arithmetic expressions. Journal of the ACM, 21(2):201-206, 1974. URL: https://doi.org/10.1145/321812.321815.
  2. Zeev Dvir, Guillaume Malod, Sylvain Perifel, and Amir Yehudayoff. Separating multilinear branching programs and formulas. In Proceedings of the 44th STOC, pages 615-624. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214034.
  3. Klim Efremenko, Ankit Garg, Rafael Oliveira, and Avi Wigderson. Barriers for rank methods in arithmetic complexity. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018), volume 94 of LIPIcs, pages 1:1-1:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.1.
  4. Wassily Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58(301):13-30, 1963. URL: https://doi.org/10.1080/01621459.1963.10500830.
  5. Pavel Hrubeš, Avi Wigderson, and Amir Yehudayoff. Non-commutative circuits and the sum-of-squares problem. Journal of the American Mathematical Society, 24(3):871-898, 2011. URL: https://doi.org/10.1090/S0894-0347-2011-00694-2.
  6. Deepanshu Kush. An unconditional barrier for proving multilinear algebraic branching program lower bounds. Electron. Colloquium Comput. Complex., TR26-043, 2026. URL: https://eccc.weizmann.ac.il/report/2026/043.
  7. Deepanshu Kush and Shubhangi Saraf. Improved low-depth set-multilinear circuit lower bounds. In 37th Computational Complexity Conference (CCC 2022), volume 234 of LIPIcs, pages 38:1-38:16, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.38.
  8. Deepanshu Kush and Shubhangi Saraf. Near-optimal set-multilinear formula lower bounds. In 38th Computational Complexity Conference (CCC 2023), volume 264 of LIPIcs, pages 15:1-15:33, 2023. URL: https://doi.org/10.4230/LIPIcs.CCC.2023.15.
  9. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. On the partial derivative method applied to lopsided set-multilinear polynomials. In Shachar Lovett, editor, 37th Computational Complexity Conference, CCC 2022, Philadelphia, PA, USA, July 20-23, 2022, volume 234 of LIPIcs, pages 32:1-32:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.32.
  10. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. Superpolynomial lower bounds against low-depth algebraic circuits. J. ACM, 72(4):26:1-26:35, 2025. URL: https://doi.org/10.1145/3734215.
  11. Noam Nisan and Avi Wigderson. Hardness vs randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994. URL: https://doi.org/10.1016/S0022-0000(05)80043-1.
  12. Noam Nisan and Avi Wigderson. Lower bounds on arithmetic circuits via partial derivatives. Comput. Complex., 6(3):217-234, 1997. URL: https://doi.org/10.1007/BF01294256.
  13. Ran Raz. Separation of multilinear circuit and formula size. Theory of Computing, 2(6):121-135, 2006. URL: https://doi.org/10.4086/TOC.2006.V002A006.
  14. Ran Raz. Multi-linear formulas for permanent and determinant are of super-polynomial size. J. ACM, 56(2):8:1-8:17, 2009. URL: https://doi.org/10.1145/1502793.1502797.
  15. Ran Raz. Tensor-rank and lower bounds for arithmetic formulas. J. ACM, 60(6):40:1-40:15, 2013. URL: https://doi.org/10.1145/2535928.
  16. Benjamin Rossman. Riffle rank. In Luciano N. Grippo and Martín D. Safe, editors, Proceedings of the XIII Latin-American Algorithms, Graphs and Optimization Symposium, LAGOS 2025, Buenos Aires, Argentina, November 10-14, 2025, volume 273 of Procedia Computer Science, pages 215-222. Elsevier, 2025. URL: https://doi.org/10.1016/j.procs.2025.10.301.
  17. Benjamin Rossman and Davidson Zhu. Multi-quadratic sum-of-squares lower bounds imply VNC ¹ ̸ = VNP. In Shubhangi Saraf, editor, 17th Innovations in Theoretical Computer Science Conference, ITCS 2026, Bocconi University, Milan, Italy, January 27-30, 2026, LIPIcs, pages 113:1-113:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026. URL: https://doi.org/10.4230/LIPIcs.ITCS.2026.113.
  18. Ramprasad Saptharishi. A survey of lower bounds in arithmetic circuit complexity, 2016. Available at URL: https://github.com/dasarpmar/lowerbounds-survey.
  19. Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5(3-4):207-388, 2010. URL: https://doi.org/10.1561/0400000039.
  20. Sébastien Tavenas, Nutan Limaye, and Srikanth Srinivasan. Set-multilinear and non-commutative formula lower bounds for iterated matrix multiplication. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 416-425. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520044.
  21. Leslie G. Valiant. Completeness classes in algebra. In Proceedings of the 11th Annual ACM Symposium on Theory of Computing (STOC), pages 249-261. ACM, 1979. URL: https://doi.org/10.1145/800135.804419.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact