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Strongly Exponential Separation between Monotone VP and Monotone VNP

Published:30 September 2020Publication History
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Abstract

We show that there is a sequence of explicit multilinear polynomials Pn (x1, … ,xn) ϵ R [x1, … ,xn] with non-negative coefficients that lies in monotone VNP such that any monotone algebraic circuit for Pn must have size exp (Ω (n)) This builds on (and strengthens) a result of Yehudayoff (STOC 2019) who showed a lower bound of exp (Ω(√n)).

References

  1. N. Alon and F. R. K. Chung. 1988. Explicit construction of linear sized tolerant networks. Discr. Math. 72, 1 (1988), 15--19. DOI:https://doi.org/10.1016/0012-365X(88)90189-6Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Allan Borodin, Alexander A. Razborov, and Roman Smolensky. 1993. On lower bounds for read-K-times branching programs. Comput. Complex. 3 (1993), 1--18. DOI:https://doi.org/10.1007/BF01200404Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Simone Bova, Florent Capelli, Stefan Mengel, and Friedrich Slivovsky. 2014. Expander CNFs have exponential DNNF size. CoRR abs/1411.1995 (2014).Google ScholarGoogle Scholar
  4. Pavol Duris, Juraj Hromkovic, Stasys Jukna, Martin Sauerhoff, and Georg Schnitger. 2004. On multi-partition communication complexity. Inf. Comput. 194, 1 (2004), 49--75. DOI:https://doi.org/10.1016/j.ic.2004.05.002Google ScholarGoogle ScholarCross RefCross Ref
  5. S. B. Gashkov. 1987. On the complexity of monotone computations of polynomials.Vestn. Mosk. Univ., Ser. I 1987, 5 (1987), 7--13.Google ScholarGoogle Scholar
  6. Sergey B. Gashkov and Igor S. Sergeev. 2012. A method for deriving lower bounds for the complexity of monotone arithmetic circuits computing real polynomials. Sbornik Math. 203, 10 (10 2012). DOI:https://doi.org/10.1070/SM2012V203N10ABEH004270Google ScholarGoogle Scholar
  7. Thomas P. Hayes. 2011. Separating the k-party communication complexity hierarchy: An application of the Zarankiewicz problem. Discr. Math. Theor. Comput. Sci. 13, 4 (2011), 15--22. Retrieved from http://dmtcs.episciences.org/546.Google ScholarGoogle Scholar
  8. Shlomo Hoory, Nathan Linial, and Avi Wigderson. 2006. Expander graphs and their applications. Bull. Amer. Math. Soc. 43, 4 (2006), 439--561.Google ScholarGoogle ScholarCross RefCross Ref
  9. Mark Jerrum and Marc Snir. 1982. Some exact complexity results for straight-line computations over semirings. J. ACM 29, 3 (1982), 874--897. DOI:https://doi.org/10.1145/322326.322341Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Stasys Jukna. 2015. Lower bounds for tropical circuits and dynamic programs. Theor. Comput. Syst. 57, 1 (2015), 160--194. DOI:https://doi.org/10.1007/s00224-014-9574-4Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. O. M. Kasim-Zade. 1986. The complexity of monotone polynomials. In Proceedings of the All-union Seminar on Discrete Mathematics and Its Applications. Moskov. Gos. Univ., Mekh.-Mat. Fak., Moscow, 136--138.Google ScholarGoogle Scholar
  12. Anup Rao and Amir Yehudayoff. 2020. Communication Complexity: and Applications. Cambridge University Press. DOI:https://doi.org/10.1017/9781108671644Google ScholarGoogle Scholar
  13. Ran Raz and Amir Yehudayoff. 2011. Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors. J. Comput. Syst. Sci. 77, 1 (2011), 167--190. DOI:https://doi.org/10.1016/j.jcss.2010.06.013Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Omer Reingold, Salil Vadhan, and Avi Wigderson. 2002. Entropy waves, the zig-zag graph product, and new constant-degree expanders. Ann. Math. 155, 1 (2002), 157--187.Google ScholarGoogle ScholarCross RefCross Ref
  15. Ramprasad Saptharishi. 2015. A survey of lower bounds in arithmetic circuit complexity. Github Survey (2015).Google ScholarGoogle Scholar
  16. C.-P. Schnorr. 1976. A lower bound on the number of additions in monotone computations. Theor. Comput. Sci. 2, 3 (1976), 305--315. DOI:https://doi.org/10.1016/0304-3975(76)90083-9Google ScholarGoogle ScholarCross RefCross Ref
  17. Eli Shamir and Marc Snir. 1977. Lower Bounds on the Number of Multiplications and the Number of Additions in Monotone Computations. IBM Thomas J. Watson Research Division.Google ScholarGoogle Scholar
  18. Amir Shpilka and Amir Yehudayoff. 2010. Arithmetic circuits: A survey of recent results and open questions. Found. Trends Theoret. Comput. Sci. 5, 3--4 (2010), 207--388. DOI:https://doi.org/10.1561/0400000039Google ScholarGoogle Scholar
  19. Leslie G. Valiant. 1979. Completeness classes in algebra. In Proceedings of the 11th Annual ACM Symposium on Theory of Computing. 249--261. DOI:https://doi.org/10.1145/800135.804419Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Leslie G. Valiant. 1980. Negation can be exponentially powerful. Theor. Comput. Sci. 12 (1980), 303--314. DOI:https://doi.org/10.1016/0304-3975(80)90060-2Google ScholarGoogle ScholarCross RefCross Ref
  21. Amir Yehudayoff. 2019. Separating monotone VP and VNP. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC’19). ACM, New York, 425--429.Google ScholarGoogle ScholarDigital LibraryDigital Library

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