research-article

Multi-linear formulas for permanent and determinant are of super-polynomial size

Author:

Ran RazAuthors Info & Claims

Journal of the ACM (JACM), Volume 56, Issue 2

Article No.: 8, Pages 1 - 17

https://doi.org/10.1145/1502793.1502797

Published: 17 April 2009 Publication History

Abstract

An arithmetic formula is multilinear if the polynomial computed by each of its subformulas is multilinear. We prove that any multilinear arithmetic formula for the permanent or the determinant of an n × n matrix is of size super-polynomial in n. Previously, super-polynomial lower bounds were not known (for any explicit function) even for the special case of multilinear formulas of constant depth.

References

[1]

Aaronson, S. 2004. Multilinear formulas and skepticism of quantum computing. In Proceedings of the Symposium on Theory of Computing (STOC 2004). ACM, New York, 118--127.

Digital Library

[2]

Alon, N., Spencer, J. H., and Erdos, P. 1992. The Probabiliatic Method. Wiley, New York.

[3]

Burgisser, P., Clausen, M., and Shokrollahi, M. A. 1997. Algebraic Complexity Theory. Springer-Verlag, New York.

[4]

Grigoriev, D., and Karpinski, M. 1998. An exponential lower bound for depth 3 arithmetic circuits. In Proceedings of the Symposium on Theory of Computing (STOC 1998). ACM, New York, 577--582.

Digital Library

[5]

Grigoriev, D., and Razborov, A. A. 2000. Exponential lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields. Appl. Algebra Eng. Commun. Comput. 10, 6, 465--487 (Preliminary version in FOCS 1998).

[6]

Impagliazzo, and R., Kabanets, V. 2004. Derandomizing polynomial identity tests means proving circuit lower bounds. Computat. Complex. 13, 1--2, 1--46 (Preliminary version in STOC 2003).

Digital Library

[7]

Kalorkoti, K. 1985. A lower bound for the formula size of rational functions. SIAM J. Comput. 14, 3, 678--687.

Digital Library

[8]

Nisan, N. 1991. Lower bounds for non-commutative computation. In Proceedings of the Symposium on Theory of Computing (STOC 1991). ACM, New York, 410--418.

Digital Library

[9]

Nisan, N., and Wigderson, A. 1996. Lower bounds on arithmetic circuits via partial derivatives. Computat. Complex. 6, 3, 217--234 (Preliminary version in FOCS 1995).

Digital Library

[10]

Raz, R. 2006. Separation of multilinear circuit and formula size. Theory of Comput. 2, 6 (Preliminary version in FOCS 2004, title: Multilinear-NC₁ ≠ Multilinear-NC₂).

[11]

Raz, R., and Shpilka, A. 2005. Deterministic polynomial identity testing in noncommutative models. Computat. Complex. 14, 1, 1--19 (Preliminary version in Conference on Computational Complexity 2004).

Digital Library

[12]

Raz, R., Shpilka, A., and Yehudayoff, A. 2007. A lower bound for the size of syntactically multilinear arithmetic circuits. In Proceedings of the Symposium on Foundations of Computer Science (FOCS 2007). IEEE Computer Society Press, Los Alamitos, CA, 438--448.

Digital Library

[13]

Raz, R., and Yehudayoff, A. 2008a. Lower bounds and separations for constant depth multilinear circuits. CCC 2008: 128--139.

Digital Library

[14]

Raz, R., and Yehudayoff, A. 2008b. Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors. In Proceedings of the Symposium on Foundations of Computer Science (FOCS 2008). IEEE Computer Society Press, Los Alamitos, CA, 273--282.

Digital Library

[15]

Schnorr, C. P. 1976. A lower bound on the number of additions in monotone computations. Theoret. Comput. Sci. 2, 3, 305--315.

[16]

Shamir, E., and Snir, M. 1980. On the depth complexity of formulas. Math. Syst. Theory 13, 301--322.

[17]

Shpilka, A., and Wigderson, A. 2001. Depth-3 arithmetic circuits over fields of characteristic zero. Computat. Complex. 10, 1, 1--27 (Preliminary version in Conference on Computational Complexity 1999).

Digital Library

[18]

Valiant, L. G. 1980. Negation can be exponentially powerful. Theoret. Comput. Sci. 12, 303--314.

[19]

Valiant, L. G. 1992. Why is Boolean complexity theory difficult&quest; In Boolean Function Complexity (M. S. Paterson, ed.) Lond. Math. Soc. Lecture Note Ser. Vol. 169, Cambridge University Press, Cambridge, UK, 84--94.

Digital Library

[20]

Von zur Gaithen, J. 1987. Feasible arithmetic computations: Valiant's hypothesis. J. Symb. Computat. 4, 2, 137--172.

Digital Library

[21]

Von zur Gaithen, J. 1988. Algebraic complexity theory. Ann. Rev. Comput. Sci. 3, 317--347.

Digital Library

Cited By

Jukna S(2025) Notes on Boolean read- and multilinear circuits Discrete Applied Mathematics10.1016/j.dam.2024.09.023360(307-327)Online publication date: Jan-2025
https://doi.org/10.1016/j.dam.2024.09.023
Wang HCong JDe Micheli G(2024)Quantum State Preparation Using an Exact CNOT Synthesis Formulation2024 Design, Automation & Test in Europe Conference & Exhibition (DATE)10.23919/DATE58400.2024.10546633(1-6)Online publication date: 25-Mar-2024
https://doi.org/10.23919/DATE58400.2024.10546633
Bhargav CDutta SSaxena N(2024)Improved Lower Bound, and Proof Barrier, for Constant Depth Algebraic CircuitsACM Transactions on Computation Theory10.1145/368995716:4(1-22)Online publication date: 11-Nov-2024
https://doi.org/10.1145/3689957
Show More Cited By

Index Terms

Multi-linear formulas for permanent and determinant are of super-polynomial size
1. Theory of computation
  1. Design and analysis of algorithms

Recommendations

Tensor-Rank and Lower Bounds for Arithmetic Formulas

We show that any explicit example for a tensor A : [n]^r → F with tensor-rank ≥ n^{rċ(1−o(1))}, where r = r(n) ≤ log n/log log n is super-constant, implies an explicit super-polynomial lower bound for the size of general arithmetic formulas over F. This ...
Multi-linear formulas for permanent and determinant are of super-polynomial size
STOC '04: Proceedings of the thirty-sixth annual ACM symposium on Theory of computing

An arithmetic formula is multi-linear if the polynomial computed by each of its sub-formulas is multi-linear. We prove that any multi-linear arithmetic formula for the permanent or the determinant of an n x n matrix is of size super-polynomial in ...
Non-commutative circuits and the sum-of-squares problem
STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

We initiate a direction for proving lower bounds on the size of non-commutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of non-commutative arithmetic circuits and a problem about commutative degree ...

Comments

Information & Contributors

Information

Published In

cover image Journal of the ACM

Journal of the ACM Volume 56, Issue 2

April 2009

190 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/1502793

Issue’s Table of Contents

Copyright © 2009 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 17 April 2009

Accepted: 01 January 2009

Revised: 01 January 2009

Received: 01 July 2004

Published in JACM Volume 56, Issue 2

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

90
Total Citations
View Citations
664
Total Downloads

Downloads (Last 12 months)41
Downloads (Last 6 weeks)2

Reflects downloads up to 20 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Jukna S(2025) Notes on Boolean read- and multilinear circuits Discrete Applied Mathematics10.1016/j.dam.2024.09.023360(307-327)Online publication date: Jan-2025
https://doi.org/10.1016/j.dam.2024.09.023
Wang HCong JDe Micheli G(2024)Quantum State Preparation Using an Exact CNOT Synthesis Formulation2024 Design, Automation & Test in Europe Conference & Exhibition (DATE)10.23919/DATE58400.2024.10546633(1-6)Online publication date: 25-Mar-2024
https://doi.org/10.23919/DATE58400.2024.10546633
Bhargav CDutta SSaxena N(2024)Improved Lower Bound, and Proof Barrier, for Constant Depth Algebraic CircuitsACM Transactions on Computation Theory10.1145/368995716:4(1-22)Online publication date: 11-Nov-2024
https://doi.org/10.1145/3689957
Fournier HLimaye NSrinivasan STavenas SMohar BShinkar IO'Donnell R(2024)On the Power of Homogeneous Algebraic FormulasProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649760(141-151)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649760
Limaye NSrinivasan STavenas S(2024)Superpolynomial Lower Bounds Against Low-Depth Algebraic CircuitsCommunications of the ACM10.1145/361109467:2(101-108)Online publication date: 25-Jan-2024
https://dl.acm.org/doi/10.1145/3611094
Bhargav CDwivedi PSaxena N(2024)Lower Bounds for the Sum of Small-Size Algebraic Branching ProgramsTheory and Applications of Models of Computation10.1007/978-981-97-2340-9_30(355-366)Online publication date: 13-May-2024
https://dl.acm.org/doi/10.1007/978-981-97-2340-9_30
Kush DSaraf S(2023)Near-Optimal Set-Multilinear Formula Lower BoundsProceedings of the conference on Proceedings of the 38th Computational Complexity Conference10.4230/LIPIcs.CCC.2023.15(1-33)Online publication date: 17-Jul-2023
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2023.15
Kush DSaraf S(2022)Improved low-depth set-multilinear circuit lower boundsProceedings of the 37th Computational Complexity Conference10.4230/LIPIcs.CCC.2022.38(1-16)Online publication date: 20-Jul-2022
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2022.38
Limaye NSrinivasan STavenas S(2022)On the partial derivative method applied to lopsided set-multilinear polynomialsProceedings of the 37th Computational Complexity Conference10.4230/LIPIcs.CCC.2022.32(1-23)Online publication date: 20-Jul-2022
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2022.32
Limaye NSrinivasan STavenas S(2022)Guest ColumnACM SIGACT News10.1145/3544979.354498953:2(40-62)Online publication date: 25-Jul-2022
https://dl.acm.org/doi/10.1145/3544979.3544989
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Issue’s Table of Contents