Lower Bounds for Special Cases of Syntactic Multilinear ABPs

  • C. Ramya
  • B. V. Raghavendra Rao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10976)


Algebraic Branching Programs (ABPs) are standard models for computing polynomials. Syntactic multilinear ABPs (smABPs) are restrictions of ABPs where every variable is allowed to occur at most once in every path from the start to terminal node. Proving lower bounds against syntactic multilinear ABPs remains a challenging open question in Algebraic Complexity Theory. The current best known bound is only quadratic [Alon,Kumar,Volk ECCC 2017].

In this article, we develop a new approach upper bounding the rank of the partial derivative matrix of syntactic multilinear ABPs: Convert the ABP to a syntactic multilinear formula with a super polynomial blow up in the size and then exploit the structural limitations of resulting formula to obtain a rank upper bound. Using this approach, we prove exponential lower bounds for special cases of smABPs and circuits namely, sum of Oblivious Read-Once ABPs, r-pass multilinear ABPs and sparse ROABPs. En route, we also prove super-polynomial lower bound for a special class of syntactic multilinear arithmetic circuits.


Computational complexity Algebraic complexity theory Algebraic branching programs 


  1. 1.
    Alon, N., Kumar, M., Volk, B.L.: An almost quadratic lower bound for syntactically multilinear arithmetic circuits. ECCC 24, 124 (2017).
  2. 2.
    Anderson, M., Forbes, M.A., Saptharishi, R., Shpilka, A., Volk, B.L.: Identity testing and lower bounds for read-k oblivious algebraic branching programs. In: CCC, pp. 30:1–30:25 (2016). Scholar
  3. 3.
    Arvind, V., Raja, S.: Some lower bound results for set-multilinear arithmetic computations. Chicago J. Theoret. Comput. Sci. (2016).
  4. 4.
    Baur, W., Strassen, V.: The complexity of partial derivatives. Theoret. Comput. Sci. 22, 317–330 (1983). Scholar
  5. 5.
    Chillara, S., Limaye, N., Srinivasan, S.: Small-depth multilinear formula lower bounds for iterated matrix multiplication, with applications. In: STACS (2018).
  6. 6.
    Forbes, M.: Polynomial identity testing of read-once oblivious algebraic branching programs. Ph.D. thesis, Massachusetts Institute of Technology (2014)Google Scholar
  7. 7.
    Jansen, M.J.: Lower bounds for syntactically multilinear algebraic branching programs. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 407–418. Springer, Heidelberg (2008). Scholar
  8. 8.
    Kayal, N., Nair, V., Saha, C.: Separation between read-once oblivious algebraic branching programs (ROABPs) and multilinear depth three circuits. In: STACS, pp. 46:1–46:15 (2016).
  9. 9.
    Mahajan, M., Tawari, A.: Sums of read-once formulas: how many summands are necessary? Theoret. Comput. Sci. 708, 34–45 (2018). Scholar
  10. 10.
    Nisan, N.: Lower bounds for non-commutative computation (extended abstract). In: STOC, pp. 410–418 (1991).
  11. 11.
    Ramya, C., Rao, B.V.R.: Sum of products of read-once formulas. In: FSTTCS, pp. 39:1–39:15 (2016).
  12. 12.
    Raz, R.: Separation of multilinear circuit and formula size. Theory Comput. 2(6), 121–135 (2006). Scholar
  13. 13.
    Raz, R.: Multi-linear formulas for permanent and determinant are of super-polynomial size. J. ACM 56(2) (2009). Scholar
  14. 14.
    Raz, R., Yehudayoff, A.: Balancing syntactically multilinear arithmetic circuits. Comput. Complex. 17(4), 515–535 (2008). Scholar
  15. 15.
    Saptharishi, R.: A survey of lower bounds in arithmetic circuit complexity (2015).
  16. 16.
    Valiant, L.G.: Completeness classes in algebra. In: STOC, pp. 249–261 (1979).

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IIT MadrasChennaiIndia

Personalised recommendations