Lower Bounds for Planar Arithmetic Circuits

Authors C. Ramya , Pratik Shastri

Thumbnail PDF


  • Filesize: 0.83 MB
  • 22 pages

Document Identifiers

Author Details

C. Ramya
  • The Institute of Mathematical Sciences (a CI of Homi Bhabha National Institute), Chennai, India
Pratik Shastri
  • The Institute of Mathematical Sciences (a CI of Homi Bhabha National Institute), Chennai, India


We thank Meena Mahajan for insightful discussions on planar arithmetic circuits. We also thank the anonymous reviewers of ITCS 2024 for their helpful comments and suggestions.

Cite AsGet BibTex

C. Ramya and Pratik Shastri. Lower Bounds for Planar Arithmetic Circuits. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 91:1-91:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Arithmetic circuits are a natural well-studied model for computing multivariate polynomials over a field. In this paper, we study planar arithmetic circuits. These are circuits whose underlying graph is planar. In particular, we prove an Ω(nlog n) lower bound on the size of planar arithmetic circuits computing explicit bilinear forms on 2n variables. As a consequence, we get an Ω(nlog n) lower bound on the size of arithmetic formulas and planar algebraic branching programs computing explicit bilinear forms. This is the first such lower bound on the formula complexity of an explicit bilinear form. In the case of read-once planar circuits, we show Ω(n²) size lower bounds for computing explicit bilinear forms. Furthermore, we prove fine separations between the various planar models of computations mentioned above. In addition to this, we look at multi-output planar circuits and show Ω(n^{4/3}) size lower bound for computing an explicit linear transformation on n-variables. For a suitable definition of multi-output formulas, we extend the above result to get an Ω(n²/log n) size lower bound. As a consequence, we demonstrate that there exists an n-variate polynomial computable by n^{1 + o(1)}-sized formulas such that any multi-output planar circuit (resp., multi-output formula) simultaneously computing all its first-order partial derivatives requires size Ω(n^{4/3}) (resp., Ω(n²/log n)). This shows that a statement analogous to that of Baur, Strassen[Walter Baur and Volker Strassen, 1983] does not hold in the case of planar circuits and formulas.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Arithmetic circuit complexity
  • Planar circuits
  • Bilinear forms


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


  1. Noga Alon and Wolfgang Maasst. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37(2):118-129, 1988. URL: https://doi.org/10.1016/0022-0000(88)90002-5.
  2. L. Babai, P. Pudlák, V. Rödl, and E. Szemeredi. Lower bounds to the complexity of symmetric boolean functions. Theoretical Computer Science, 74(3):313-323, 1990. URL: https://doi.org/10.1016/0304-3975(90)90080-2.
  3. Walter Baur and Volker Strassen. The complexity of partial derivatives. Theoretical Computer Science, 22(3):317-330, 1983. URL: https://doi.org/10.1016/0304-3975(83)90110-X.
  4. Sandeep Bhatt, Fan Chung, Tom Leighton, and Arnold Rosenberg. Optimal simulations of tree machines. In 27th Annual Symposium on Foundations of Computer Science (sfcs 1986), pages 274-282, 1986. URL: https://doi.org/10.1109/SFCS.1986.38.
  5. Sandeep N. Bhatt and Charles E. Leiserson. How to assemble tree machines (extended abstract). In Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing, STOC '82, pages 77-84, New York, NY, USA, 1982. Association for Computing Machinery. URL: https://doi.org/10.1145/800070.802179.
  6. Abhranil Chatterjee, Mrinal Kumar, and Ben Lee Volk. Determinants vs. algebraic branching programs. Electron. Colloquium Comput. Complex., TR23-115, 2023. URL: https://arxiv.org/abs/TR23-115.
  7. Prerona Chatterjee, Mrinal Kumar, Adrian She, and Ben Lee Volk. Quadratic lower bounds for algebraic branching programs and formulas. computational complexity, 31(2):8, July 2022. URL: https://doi.org/10.1007/s00037-022-00223-8.
  8. Xi Chen, Neeraj Kayal, and Avi Wigderson. Partial derivatives in arithmetic complexity and beyond. Foundations and Trends in Theoretical Computer Science, January 2010. URL: https://www.microsoft.com/en-us/research/publication/partial-derivatives-in-arithmetic-complexity-and-beyond/.
  9. Hans Dietmar Gröger. A new partition lemma for planar graphs and its application to circuit complexity. In Lothar Budach, editor, Fundamentals of Computation Theory, 8th International Symposium, FCT '91, Gosen, Germany, September 9-13, 1991, Proceedings, volume 529 of Lecture Notes in Computer Science, pages 220-229. Springer, 1991. URL: https://doi.org/10.1007/3-540-54458-5_66.
  10. K. A. Kalorkoti. A lower bound for the formula size of rational functions. SIAM Journal on Computing, 14(3):678-687, 1985. URL: https://doi.org/10.1137/0214050.
  11. Erich Kaltofen and Michael F. Singer. Size efficient parallel algebraic circuits for partial derivatives. In IV International Conference on Computer Algebra in Physical Research, 1990. URL: https://users.cs.duke.edu/~elk27/bibliography/91/KaSi91.pdf.
  12. Richard J. Lipton and Robert Endre Tarjan. Applications of a planar separator theorem. In 18th Annual Symposium on Foundations of Computer Science (sfcs 1977), pages 162-170, 1977. URL: https://doi.org/10.1109/SFCS.1977.6.
  13. Richard J. Lipton and Robert Endre Tarjan. A separator theorem for planar graphs. SIAM Journal on Applied Mathematics, 36(2):177-189, 1979. URL: https://doi.org/10.1137/0136016.
  14. Noam Nisan and Mario Szegedy. On the degree of boolean functions as real polynomials. computational complexity, 4(4):301-313, December 1994. URL: https://doi.org/10.1007/BF01263419.
  15. Noam Nisan and Avi Wigderson. On the complexity of bilinear forms: Dedicated to the memory of jacques morgenstern. In Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, STOC '95, pages 723-732, New York, NY, USA, 1995. Association for Computing Machinery. URL: https://doi.org/10.1145/225058.225290.
  16. Jaikumar Radhakrishnan and Amnon Ta-Shma. Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM Journal on Discrete Mathematics, 13(1):2-24, 2000. URL: https://doi.org/10.1137/S0895480197329508.
  17. Ran Raz. On the complexity of matrix product. In Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, STOC '02, pages 144-151, New York, NY, USA, 2002. Association for Computing Machinery. URL: https://doi.org/10.1145/509907.509932.
  18. Ran Raz. Elusive functions and lower bounds for arithmetic circuits. Theory of Computing, 6(7):135-177, 2010. URL: https://doi.org/10.4086/toc.2010.v006a007.
  19. John E. Savage. Planar Circuit Complexity and The Performance of VLSI Algorithms +, pages 61-68. Springer Berlin Heidelberg, Berlin, Heidelberg, 1981. URL: https://doi.org/10.1007/978-3-642-68402-9_8.
  20. John E. Savage. The performance of multilective vlsi algorithms. Journal of Computer and System Sciences, 29(2):243-273, 1984. URL: https://doi.org/10.1016/0022-0000(84)90033-3.
  21. Amnon Ta-Shma. On extracting randomness from weak random sources (extended abstract). In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC '96, pages 276-285, New York, NY, USA, 1996. Association for Computing Machinery. URL: https://doi.org/10.1145/237814.237877.
  22. György Turán. On the complexity of planar boolean circuits. computational complexity, 5(1):24-42, March 1995. URL: https://doi.org/10.1007/BF01277954.
  23. Leslie G. Valiant. On non-linear lower bounds in computational complexity. In Proceedings of the Seventh Annual ACM Symposium on Theory of Computing, STOC '75, pages 45-53, New York, NY, USA, 1975. Association for Computing Machinery. URL: https://doi.org/10.1145/800116.803752.
  24. Leslie G. Valiant. Graph-theoretic arguments in low-level complexity. In Jozef Gruska, editor, Mathematical Foundations of Computer Science 1977, pages 162-176, Berlin, Heidelberg, 1977. Springer Berlin Heidelberg. Google Scholar