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Lower bounds for multilinear bounded order ABPs
Proving super-polynomial size lower bounds for syntactic multilinear
Algebraic Branching Programs(smABPs) computing an explicit polynomial is a
challenging problem in Algebraic Complexity Theory. The order in which
variables in appear along source to sink paths in any
smABP can be viewed as a permutation in . 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:
Strict circular-interval ABPs: For every subprogram the index set of
variables occurring in it is contained in some circular interval of
.
L-ordered ABPs: There is a set of L permutations of variables such that every
source to sink path in the ABP reads variables in one of the L orders.
We prove exponential lower bound for the size of a strict circular-interval
ABP computing an explicit n-variate multilinear polynomial in VP. For the same
polynomial, we show that any sum of L-ordered ABPs of small size will require
exponential () many summands, when . At the heart of above lower bound arguments
is a new decomposition theorem for smABPs: We show that any polynomial
computable 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 obtain a low
bottom fan-in version of the depth reduction by Tavenas [MFCS 2013] in the case
of smABPs. In particular, we show that a polynomial having size S smABPs can be
expressed as a sum of products of multilinear polynomials on
variables, where the total number of summands is bounded by . Additionally, we show that L-ordered ABPs can be transformed into
L-pass smABPs with a polynomial blowup in size