License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.MFCS.2019.52
URN: urn:nbn:de:0030-drops-109963
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Ramya, C. ; Rao, B. V. Raghavendra

Lower Bounds for Multilinear Order-Restricted ABPs

LIPIcs-MFCS-2019-52.pdf (0.6 MB)


Proving super-polynomial lower bounds on the size of syntactic multilinear Algebraic Branching Programs (smABPs) computing an explicit polynomial is a challenging problem in Algebraic Complexity Theory. The order in which variables in {x_1,...,x_n} appear along any source to sink path in an smABP can be viewed as a permutation in S_n. In this article, we consider the following special classes of smABPs where the order of occurrence of variables along a source to sink path is restricted:
1) Strict circular-interval ABPs: For every sub-program the index set of variables occurring in it is contained in some circular interval of {1,..., n}.
2) L-ordered ABPs: There is a set of L permutations (orders) of variables such that every source to sink path in the smABP reads variables in one of these L orders, where L <=2^{n^{1/2 -epsilon}} for some epsilon>0.
We prove exponential (i.e., 2^{Omega(n^delta)}, delta>0) lower bounds on the size of above models computing an explicit multilinear 2n-variate polynomial in VP.
As a main ingredient in our lower bounds, we show that any polynomial that can be computed by an smABP of size S, can be written as a sum of O(S) many multilinear polynomials where each summand is a product of two polynomials in at most 2n/3 variables, computable by smABPs. As a corollary, we show that any size S syntactic multilinear ABP can be transformed into a size S^{O(sqrt{n})} depth four syntactic multilinear Sigma Pi Sigma Pi circuit where the bottom Sigma gates compute polynomials on at most O(sqrt{n}) variables.
Finally, we compare the above models with other standard models for computing multilinear polynomials.

BibTeX - Entry

  author =	{C. Ramya and B. V. Raghavendra Rao},
  title =	{{Lower Bounds for Multilinear Order-Restricted ABPs}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{52:1--52:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Peter Rossmanith and Pinar Heggernes and Joost-Pieter Katoen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-109963},
  doi =		{10.4230/LIPIcs.MFCS.2019.52},
  annote =	{Keywords: Computational complexity, Algebraic complexity theory, Polynomials}

Keywords: Computational complexity, Algebraic complexity theory, Polynomials
Collection: 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)
Issue Date: 2019
Date of publication: 20.08.2019

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