Document Open Access Logo

Near-Optimal Set-Multilinear Formula Lower Bounds

Authors Deepanshu Kush , Shubhangi Saraf

Thumbnail PDF


  • Filesize: 0.92 MB
  • 33 pages

Document Identifiers

Author Details

Deepanshu Kush
  • Department of Computer Science, University of Toronto, Canada
Shubhangi Saraf
  • Department of Computer Science and Department of Mathematics, University of Toronto, Canada


We would like to thank Srikanth Srinivasan for several helpful and insightful discussions. Our early discussions with him were what inspired much of this work.

Cite AsGet BibTex

Deepanshu Kush and Shubhangi Saraf. Near-Optimal Set-Multilinear Formula Lower Bounds. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 15:1-15:33, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


The seminal work of Raz (J. ACM 2013) as well as the recent breakthrough results by Limaye, Srinivasan, and Tavenas (FOCS 2021, STOC 2022) have demonstrated a potential avenue for obtaining lower bounds for general algebraic formulas, via strong enough lower bounds for set-multilinear formulas. In this paper, we make progress along this direction by proving near-optimal lower bounds against low-depth as well as unbounded-depth set-multilinear formulas. More precisely, we show that over any field of characteristic zero, there is a polynomial f computed by a polynomial-sized set-multilinear branching program (i.e., f is in set-multilinear VBP) defined over Θ(n²) variables and of degree Θ(n), such that any product-depth Δ set-multilinear formula computing f has size at least n^Ω(n^{1/Δ}/Δ). Moreover, we show that any unbounded-depth set-multilinear formula computing f has size at least n^{Ω(log n)}. If such strong lower bounds are proven for the iterated matrix multiplication (IMM) polynomial or rather, any polynomial that is computed by an ordered set-multilinear branching program (i.e., a further restriction of set-multilinear VBP), then this would have dramatic consequences as it would imply super-polynomial lower bounds for general algebraic formulas (Raz, J. ACM 2013; Tavenas, Limaye, and Srinivasan, STOC 2022). Prior to our work, either only weaker lower bounds were known for the IMM polynomial (Tavenas, Limaye, and Srinivasan, STOC 2022), or similar strong lower bounds were known but for a hard polynomial not known to be even in set-multilinear VP (Kush and Saraf, CCC 2022; Raz, J. ACM 2009). By known depth-reduction results, our lower bounds are essentially tight for f and in general, for any hard polynomial that is in set-multilinear VBP or set-multilinear VP. Any asymptotic improvement in the lower bound (for a hard polynomial, say, in VNP) would imply super-polynomial lower bounds for general set-multilinear circuits.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Algebraic Complexity
  • Set-multilinear
  • Formula Lower Bounds


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


  1. Manindra Agrawal, Rohit Gurjar, Arpita Korwar, and Nitin Saxena. Hitting-sets for ROABP and sum of set-multilinear circuits. SIAM J. Comput., 44(3):669-697, 2015. URL:
  2. Manindra Agrawal and V. Vinay. Arithmetic circuits: A chasm at depth four. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 67-75. IEEE Computer Society, 2008. URL:
  3. Miklós Ajtai. ∑^1_1-formulae on finite structures. Ann. Pure Appl. Log., 24(1):1-48, 1983. URL:
  4. Vikraman Arvind and S. Raja. Some lower bound results for set-multilinear arithmetic computations. Chic. J. Theor. Comput. Sci., 2016, 2016. URL:
  5. Peter Bürgisser. Cook’s versus valiant’s hypothesis. Theor. Comput. Sci., 235(1):71-88, 2000. URL:
  6. Suryajith Chillara, Christian Engels, Nutan Limaye, and Srikanth Srinivasan. A near-optimal depth-hierarchy theorem for small-depth multilinear circuits. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 934-945. IEEE Computer Society, 2018. URL:
  7. Suryajith Chillara, Nutan Limaye, and Srikanth Srinivasan. Small-depth multilinear formula lower bounds for iterated matrix multiplication with applications. SIAM J. Comput., 48(1):70-92, 2019. URL:
  8. Zeev Dvir, Guillaume Malod, Sylvain Perifel, and Amir Yehudayoff. Separating multilinear branching programs and formulas. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19-22, 2012, pages 615-624. ACM, 2012. URL:
  9. Michael A. Forbes and Amir Shpilka. Quasipolynomial-time identity testing of non-commutative and read-once oblivious algebraic branching programs. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 243-252. IEEE Computer Society, 2013. URL:
  10. Hervé Fournier, Nutan Limaye, Guillaume Malod, and Srikanth Srinivasan. Lower bounds for depth-4 formulas computing iterated matrix multiplication. SIAM J. Comput., 44(5):1173-1201, 2015. URL:
  11. Merrick L. Furst, James B. Saxe, and Michael Sipser. Parity, circuits, and the polynomial-time hierarchy. Math. Syst. Theory, 17(1):13-27, 1984. URL:
  12. Zeyu Guo and Rohit Gurjar. Improved explicit hitting-sets for roabps. In Jaroslaw Byrka and Raghu Meka, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2020, August 17-19, 2020, Virtual Conference, volume 176 of LIPIcs, pages 4:1-4:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  13. Johan Håstad. Almost optimal lower bounds for small depth circuits. In Juris Hartmanis, editor, Proceedings of the 18th Annual ACM Symposium on Theory of Computing, May 28-30, 1986, Berkeley, California, USA, pages 6-20. ACM, 1986. URL:
  14. Pavel Hrubes and Amir Yehudayoff. Homogeneous formulas and symmetric polynomials. Comput. Complex., 20(3):559-578, 2011. URL:
  15. Neeraj Kayal, Nutan Limaye, Chandan Saha, and Srikanth Srinivasan. An exponential lower bound for homogeneous depth four arithmetic formulas. SIAM J. Comput., 46(1):307-335, 2017. URL:
  16. Neeraj Kayal, Vineet Nair, and Chandan Saha. Separation between read-once oblivious algebraic branching programs (roabps) and multilinear depth-three circuits. ACM Trans. Comput. Theory, 12(1):2:1-2:27, 2020. URL:
  17. Neeraj Kayal, Chandan Saha, and Ramprasad Saptharishi. A super-polynomial lower bound for regular arithmetic formulas. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 146-153. ACM, 2014. URL:
  18. Neeraj Kayal, Chandan Saha, and Sébastien Tavenas. An almost cubic lower bound for depth three arithmetic circuits. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 33:1-33:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL:
  19. Neeraj Kayal, Chandan Saha, and Sébastien Tavenas. On the size of homogeneous and of depth-four formulas with low individual degree. Theory Comput., 14(1):1-46, 2018. URL:
  20. Pascal Koiran. Arithmetic circuits: The chasm at depth four gets wider. Theor. Comput. Sci., 448:56-65, 2012. URL:
  21. Mrinal Kumar and Shubhangi Saraf. On the power of homogeneous depth 4 arithmetic circuits. SIAM J. Comput., 46(1):336-387, 2017. URL:
  22. Deepanshu Kush and Shubhangi Saraf. Improved low-depth set-multilinear circuit lower bounds. In Shachar Lovett, editor, 37th Computational Complexity Conference, CCC 2022, July 20-23, 2022, Philadelphia, PA, USA, volume 234 of LIPIcs, pages 38:1-38:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL:
  23. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. Superpolynomial lower bounds against low-depth algebraic circuits. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 804-814. IEEE, 2021. URL:
  24. Noam Nisan and Avi Wigderson. Hardness vs randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994. URL:
  25. Noam Nisan and Avi Wigderson. Lower bounds on arithmetic circuits via partial derivatives. Comput. Complex., 6(3):217-234, 1997. URL:
  26. Ran Raz. Separation of multilinear circuit and formula size. Theory Comput., 2(6):121-135, 2006. URL:
  27. Ran Raz. Multi-linear formulas for permanent and determinant are of super-polynomial size. J. ACM, 56(2):8:1-8:17, 2009. URL:
  28. Ran Raz. Tensor-rank and lower bounds for arithmetic formulas. J. ACM, 60(6):40:1-40:15, 2013. URL:
  29. Ran Raz and Amir Yehudayoff. Balancing syntactically multilinear arithmetic circuits. Comput. Complex., 17(4):515-535, 2008. URL:
  30. Ran Raz and Amir Yehudayoff. Lower bounds and separations for constant depth multilinear circuits. Comput. Complex., 18(2):171-207, 2009. URL:
  31. Alexander A. Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mathematical notes of the Academy of Sciences of the USSR, 41:333-338, 1987. Google Scholar
  32. Ramprasad Saptharishi. A survey of lower bounds in arithmetic circuit complexity. Github Survey, 2015. URL:
  33. Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Found. Trends Theor. Comput. Sci., 5(3-4):207-388, 2010. URL:
  34. Roman Smolensky. Algebraic methods in the theory of lower bounds for boolean circuit complexity. In Alfred V. Aho, editor, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, New York, New York, USA, pages 77-82. ACM, 1987. URL:
  35. Sébastien Tavenas. Improved bounds for reduction to depth 4 and depth 3. Inf. Comput., 240:2-11, 2015. URL:
  36. 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:
  37. Leslie G. Valiant, Sven Skyum, Stuart J. Berkowitz, and Charles Rackoff. Fast parallel computation of polynomials using few processors. SIAM J. Comput., 12:641-644, 1983. Google Scholar
  38. Andrew Chi-Chih Yao. Separating the polynomial-time hierarchy by oracles (preliminary version). In 26th Annual Symposium on Foundations of Computer Science, Portland, Oregon, USA, 21-23 October 1985, pages 1-10. IEEE Computer Society, 1985. 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