Document Open Access Logo

Nonnegative Rank Measures and Monotone Algebraic Branching Programs

Authors Hervé Fournier, Guillaume Malod, Maud Szusterman, Sébastien Tavenas

PDF

File

Thumbnail PDF
PDF
LIPIcs.FSTTCS.2019.15.pdf
  • Filesize: 493 kB
  • 14 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Complexity classes
Keywords
  • Elementary symmetric polynomials
  • lower bounds

Metrics

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

Abstract

Inspired by Nisan’s characterization of noncommutative complexity (Nisan 1991), we study different notions of nonnegative rank, associated complexity measures and their link with monotone computations. In particular we answer negatively an open question of Nisan asking whether nonnegative rank characterizes monotone noncommutative complexity for algebraic branching programs. We also prove a rather tight lower bound for the computation of elementary symmetric polynomials by algebraic branching programs in the monotone setting or, equivalently, in the homogeneous syntactically multilinear setting.

Cite As Get BibTex

Hervé Fournier, Guillaume Malod, Maud Szusterman, and Sébastien Tavenas. Nonnegative Rank Measures and Monotone Algebraic Branching Programs. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.FSTTCS.2019.15

Author Details

Hervé Fournier
  • Université de Paris, IMJ-PRG, CNRS, F-75013 Paris, France
Guillaume Malod
  • Université de Paris, IMJ-PRG, CNRS, F-75013 Paris, France
Maud Szusterman
  • Université de Paris, IMJ-PRG, CNRS, F-75013 Paris, France
Sébastien Tavenas
  • Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France

Funding

  • Tavenas, Sébastien: This work was partially funded by the project BIRCA from the IDEX Université Grenoble Alpes.

References

  1. 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, June 22-24, 2018, San Diego, CA, USA, pages 11:1-11:16, 2018. URL: https://doi.org/10.4230/LIPIcs.CCC.2018.11.
  2. 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.
  3. LeRoy B. Beasley and Thomas J. Laffey. Real rank versus nonnegative rank. Linear Algebra Appl., 431(12):2330-2335, 2009. URL: https://doi.org/10.1016/j.laa.2009.02.034.
  4. Symeon Bozapalidis and Olympia Louscou-Bozapalidou. The rank of a formal tree power series. Theoretical Computer Science, 27(1):211-215, 1983. URL: https://doi.org/10.1016/0304-3975(83)90100-7.
  5. Joel E Cohen and Uriel G Rothblum. Nonnegative ranks, decompositions, and factorizations of nonnegative matrices. Linear Algebra and its Applications, 190:149-168, 1993. Google Scholar
  6. Ronald de Wolf. Nondeterministic Quantum Query and Communication Complexities. SIAM J. Comput., 32(3):681-699, 2003. URL: https://doi.org/10.1137/S0097539702407345.
  7. Nathanaël Fijalkow, Guillaume Lagarde, Pierre Ohlmann, and Olivier Serre. Lower bounds for arithmetic circuits via the Hankel matrix. Electronic Colloquium on Computational Complexity (ECCC), 25:180, 2018. URL: https://eccc.weizmann.ac.il/report/2018/180.
  8. Samuel Fiorini, Serge Massar, Sebastian Pokutta, Hans Raj Tiwary, and Ronald de Wolf. Exponential Lower Bounds for Polytopes in Combinatorial Optimization. J. ACM, 62(2):17:1-17:23, 2015. URL: https://doi.org/10.1145/2716307.
  9. Samuel Fiorini, Thomas Rothvoß, and Hans Raj Tiwary. Extended Formulations for Polygons. Discrete & Computational Geometry, 48(3):658-668, 2012. URL: https://doi.org/10.1007/s00454-012-9421-9.
  10. Michel Fliess. Matrices de Hankel. J. Math. Pures Appl. (9), 53:197-222, 1974. Google Scholar
  11. Hervé Fournier, Guillaume Malod, Maud Szusterman, and Sébastien Tavenas. Nonnegative rank measures and monotone algebraic branching programs. Electronic Colloquium on Computational Complexity (ECCC), 26:100, 2019. URL: https://eccc.weizmann.ac.il/report/2019/100.
  12. Peter Frankl and Vojtěch Rödl. Forbidden intersections. Trans. Amer. Math. Soc., 300(1):259-286, 1987. URL: https://doi.org/10.2307/2000598.
  13. Pavel Hrubes and Amir Yehudayoff. Homogeneous Formulas and Symmetric Polynomials. Computational Complexity, 20(3):559-578, 2011. URL: https://doi.org/10.1007/s00037-011-0007-3.
  14. Pavel Hrubes and Amir Yehudayoff. Formulas are Exponentially Stronger than Monotone Circuits in Non-commutative Setting. In Proceedings of the 28th Conference on Computational Complexity, CCC 2013, K.lo Alto, California, USA, 5-7 June, 2013, pages 10-14, 2013. URL: https://doi.org/10.1109/CCC.2013.11.
  15. Mark Jerrum and Marc Snir. Some Exact Complexity Results for Straight-Line Computations over Semirings. J. ACM, 29(3):874-897, July 1982. URL: https://doi.org/10.1145/322326.322341.
  16. Stasys Jukna and Georg Schnitger. On the optimality of Bellman-Ford-Moore shortest path algorithm. Theor. Comput. Sci., 628:101-109, 2016. URL: https://doi.org/10.1016/j.tcs.2016.03.014.
  17. Adam Klivans and Amir Shpilka. Learning Restricted Models of Arithmetic Circuits. Theory of Computing, 2(10):185-206, 2006. URL: https://doi.org/10.4086/toc.2006.v002a010.
  18. Mrinal Kumar and Shubhangi Saraf. On the Power of Homogeneous Depth 4 Arithmetic Circuits. SIAM J. Comput., 46(1):336-387, 2017. URL: https://doi.org/10.1137/140999335.
  19. Noam Nisan. Lower Bounds for Non-Commutative Computation (Extended Abstract). In Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, May 5-8, 1991, New Orleans, Louisiana, USA, pages 410-418, 1991. URL: https://doi.org/10.1145/103418.103462.
  20. Noam Nisan and Avi Wigderson. Lower bounds on arithmetic circuits via partial derivatives. Comput. Complexity, 6(3):217-234, 1996/97. URL: https://doi.org/10.1007/BF01294256.
  21. 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.
  22. C.P. Schnorr. A lower bound on the number of additions in monotone computations. Theoretical Computer Science, 2(3):305-315, 1976. URL: https://doi.org/10.1016/0304-3975(76)90083-9.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact