Document Open Access Logo

Unbalancing Sets and an Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits

Authors Noga Alon, Mrinal Kumar, Ben Lee Volk

PDF
Thumbnail PDF

File

LIPIcs.CCC.2018.11.pdf
  • Filesize: 0.5 MB
  • 16 pages

Document Identifiers

Author Details

Noga Alon
  • Sackler School of Mathematics and Blavatnik School of Computer Science , Tel Aviv, 6997801, Israel, and , Center for Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA
Mrinal Kumar
  • Center for Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA
Ben Lee Volk
  • Blavatnik School of Computer Science, Tel Aviv University , Tel Aviv, 6997801, Israel

Funding

  • Alon, Noga: Research supported in part by an ISF grant and by a GIF grant.
  • Kumar, Mrinal: Part of this work was done while visiting Tel Aviv University.
  • Volk, Ben Lee: The research leading to these results has received funding from the Israel Science Foundation (grant number 552/16).

Cite As Get BibTex

Noga Alon, Mrinal Kumar, and Ben Lee Volk. Unbalancing Sets and an Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.CCC.2018.11

Abstract

We prove a lower bound of Omega(n^2/log^2 n) on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial f(x_1, ..., x_n). Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff ([Ran Raz et al., 2008]), who proved a lower bound of Omega(n^{4/3}/log^2 n) for the same polynomial. Our improvement follows from an asymptotically optimal lower bound for a generalized version of Galvin's problem in extremal set theory.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Algebraic Complexity
  • Multilinear Circuits
  • Circuit Lower Bounds

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Noga Alon, E. E. Bergmann, Don Coppersmith, and Andrew M. Odlyzko. Balancing sets of vectors. IEEE Trans. Information Theory, 34(1):128-130, 1988. URL: http://dx.doi.org/10.1109/18.2610.
  2. Noga Alon, Mrinal Kumar, and Ben Lee Volk. An almost quadratic lower bound for syntactically multilinear arithmetic circuits. Electronic Colloquium on Computational Complexity (ECCC), 24:124, 2017. URL: https://eccc.weizmann.ac.il/report/2017/124.
  3. Noga Alon and Joel H. Spencer. The Probabilistic Method. Wiley Publishing, 4th edition, 2016. URL: http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1119061954.html.
  4. Richard P. Anstee, Lajos Rónyai, and Attila Sali. Shattering news. Graphs and Combinatorics, 18(1):59-73, 2002. URL: http://dx.doi.org/10.1007/s003730200003.
  5. Walter Baur and Volker Strassen. The complexity of partial derivatives. Theor. Comput. Sci., 22:317-330, 1983. URL: http://dx.doi.org/10.1016/0304-3975(83)90110-X.
  6. Stuart J. Berkowitz. On computing the determinant in small parallel time using a small number of processors. Inf. Process. Lett., 18(3):147-150, 1984. URL: http://dx.doi.org/10.1016/0020-0190(84)90018-8.
  7. Xi Chen, Neeraj Kayal, and Avi Wigderson. Partial derivatives in arithmetic complexity and beyond. Foundations and Trends in Theoretical Computer Science, 6(1-2):1-138, 2011. URL: http://dx.doi.org/10.1561/0400000043.
  8. L. Csanky. Fast parallel matrix inversion algorithms. SIAM J. Comput., 5(4):618-623, 1976. URL: http://dx.doi.org/10.1137/0205040.
  9. H. Enomoto, Peter Frankl, N. Ito, and K. Nomura. Codes with given distances. Graphs and Combinatorics, 3(1):25-38, 1987. URL: http://dx.doi.org/10.1007/BF01788526.
  10. Jeffrey B. Farr and Shuhong Gao. Computing gröbner bases for vanishing ideals of finite sets of points. In Marc P. C. Fossorier, Hideki Imai, Shu Lin, and Alain Poli, editors, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 16th International Symposium, AAECC-16, Las Vegas, NV, USA, February 20-24, 2006, Proceedings, volume 3857 of Lecture Notes in Computer Science, pages 118-127. Springer, 2006. URL: http://dx.doi.org/10.1007/11617983_11.
  11. Hervé Fournier, Nutan Limaye, Guillaume Malod, and Srikanth Srinivasan. Lower bounds for depth 4 formulas computing iterated matrix multiplication. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 128-135. ACM, 2014. URL: http://dx.doi.org/10.1145/2591796.2591824.
  12. Peter Frankl and Vojtěch Rödl. Forbidden intersections. Trans. Amer. Math. Soc., 300(1):259-286, 1987. URL: http://dx.doi.org/10.2307/2000598.
  13. Dima Grigoriev and Marek Karpinski. An exponential lower bound for depth 3 arithmetic circuits. In Jeffrey Scott Vitter, editor, Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages 577-582. ACM, 1998. URL: http://dx.doi.org/10.1145/276698.276872.
  14. Dima Grigoriev and Alexander A. Razborov. Exponential lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields. Appl. Algebra Eng. Commun. Comput., 10(6):465-487, 2000. URL: http://dx.doi.org/10.1007/s002009900021.
  15. Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Approaching the chasm at depth four. J. ACM, 61(6):33:1-33:16, 2014. URL: http://dx.doi.org/10.1145/2629541.
  16. Gábor Hegedűs. Balancing sets of vectors. Studia Sci. Math. Hungar., 47(3):333-349, 2010. URL: http://dx.doi.org/10.1556/SScMath.2009.1134.
  17. Gábor Hegedűs and Lajos Rónyai. Gröbner bases for complete uniform families. J. Algebraic Combin., 17(2):171-180, 2003. URL: http://dx.doi.org/10.1023/A:1022934815185.
  18. Maurice J. Jansen. Lower bounds for syntactically multilinear algebraic branching programs. In Edward Ochmanski and Jerzy Tyszkiewicz, editors, Mathematical Foundations of Computer Science 2008, 33rd International Symposium, MFCS 2008, Torun, Poland, August 25-29, 2008, Proceedings, volume 5162 of Lecture Notes in Computer Science, pages 407-418. Springer, 2008. URL: http://dx.doi.org/10.1007/978-3-540-85238-4_33.
  19. K. Kalorkoti. A lower bound for the formula size of rational functions. SIAM J. Comput., 14(3):678-687, 1985. URL: http://dx.doi.org/10.1137/0214050.
  20. Neeraj Kayal, Nutan Limaye, Chandan Saha, and Srikanth Srinivasan. An exponential lower bound for homogeneous depth four arithmetic formulas. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 61-70. IEEE Computer Society, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.15.
  21. Donald E. Knuth. Efficient balanced codes. IEEE Trans. Information Theory, 32(1):51-53, 1986. URL: http://dx.doi.org/10.1109/TIT.1986.1057136.
  22. Mrinal Kumar. A quadratic lower bound for homogeneous algebraic branching programs. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages 19:1-19:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.19.
  23. Mrinal Kumar and Ramprasad Saptharishi. An exponential lower bound for homogeneous depth-5 circuits over finite fields. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages 31:1-31:30. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.31.
  24. Mrinal Kumar and Shubhangi Saraf. On the power of homogeneous depth 4 arithmetic circuits. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 364-373. IEEE Computer Society, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.46.
  25. Meena Mahajan and V. Vinay. Determinant: Combinatorics, algorithms, and complexity. Chicago J. Theor. Comput. Sci., 1997, 1997. Preliminary version in the \nth\intcalcSub19971989 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1997). URL: http://cjtcs.cs.uchicago.edu/articles/1997/5/contents.html.
  26. Noam Nisan. Lower bounds for non-commutative computation (extended abstract). In Cris Koutsougeras and Jeffrey Scott Vitter, editors, Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, May 5-8, 1991, New Orleans, Louisiana, USA, pages 410-418. ACM, 1991. URL: http://dx.doi.org/10.1145/103418.103462.
  27. Noam Nisan and Avi Wigderson. Lower bounds on arithmetic circuits via partial derivatives. Computational Complexity, 6(3):217-234, 1997. URL: http://dx.doi.org/10.1007/BF01294256.
  28. Ran Raz. Separation of multilinear circuit and formula size. Theory of Computing, 2(6):121-135, 2006. URL: http://dx.doi.org/10.4086/toc.2006.v002a006.
  29. Ran Raz. Multi-linear formulas for permanent and determinant are of super-polynomial size. J. ACM, 56(2):8:1-8:17, 2009. URL: http://dx.doi.org/10.1145/1502793.1502797.
  30. Ran Raz. Elusive functions and lower bounds for arithmetic circuits. Theory of Computing, 6(1):135-177, 2010. URL: http://dx.doi.org/10.4086/toc.2010.v006a007.
  31. Ran Raz, Amir Shpilka, and Amir Yehudayoff. A lower bound for the size of syntactically multilinear arithmetic circuits. SIAM J. Comput., 38(4):1624-1647, 2008. URL: http://dx.doi.org/10.1137/070707932.
  32. Ran Raz and Amir Yehudayoff. Balancing syntactically multilinear arithmetic circuits. Computational Complexity, 17(4):515-535, 2008. URL: http://dx.doi.org/10.1007/s00037-008-0254-0.
  33. Ran Raz and Amir Yehudayoff. Lower bounds and separations for constant depth multilinear circuits. Computational Complexity, 18(2):171-207, 2009. URL: http://dx.doi.org/10.1007/s00037-009-0270-8.
  34. Ramprasad Saptharishi. A survey of lower bounds in arithmetic circuit complexity. Github survey, https://github.com/dasarpmar/lowerbounds-survey/, 2016. URL: https://github.com/dasarpmar/lowerbounds-survey/releases/.
  35. 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: http://dx.doi.org/10.1561/0400000039.
  36. Matthew Skala. Hypergeometric tail inequalities: ending the insanity. arXiv preprint arXiv:1311.5939, 2013. URL: https://arxiv.org/abs/1311.5939.
  37. Volker Strassen. Die berechnungskomplexität von elementarsymmetrischen funktionen und von interpolationskoeffizienten. Numerische Mathematik, 20(3):238-251, 1973. URL: http://dx.doi.org/10.1007/BF01436566.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact