Lower Bounds for Special Cases of Syntactic Multilinear ABPs

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 the 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 multlinear ABPs: Convert the ABP to a syntactic mulilinear 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 mulitlinear ABPs and sparse ROABPs. En route, we also prove super-polynomial lower bound for a special class of syntactic multilinear arithmetic circuits

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 $\{x_1,\ldots,x_n\}$ appear along source to sink paths in any 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: Strict circular-interval ABPs: For every subprogram the index set of variables occurring in it is contained in some circular interval of $\{1,\ldots,n\}$. 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 ($2^{n^{\Omega(1)}}$) many summands, when $L \leq 2^{n^{1/2-\epsilon}}, \epsilon>0$. 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 $O(\sqrt{n})$ variables, where the total number of summands is bounded by $2^{O(\sqrt{n}\log n \log S)}$. Additionally, we show that L-ordered ABPs can be transformed into L-pass smABPs with a polynomial blowup in size

Limitations of Sums of Bounded-Read Formulas

Proving super polynomial size lower bounds for various classes of arithmetic circuits computing explicit polynomials is a very important and challenging task in algebraic complexity theory. We study representation of polynomials as sums of weaker models such as read once formulas (ROFs) and read once oblivious algebraic branching programs (ROABPs). We prove: (1) An exponential separation between sum of ROFs and read-$k$ formulas for some constant $k$. (2) A sub-exponential separation between sum of ROABPs and syntactic multilinear ABPs. Our results are based on analysis of the partial derivative matrix under different distributions. These results highlight richness of bounded read restrictions in arithmetic formulas and ABPs. Finally, we consider a generalization of multilinear ROABPs known as strict-interval ABPs defined in [Ramya-Rao, MFCS2019]. We show that strict-interval ABPs are equivalent to ROABPs upto a polynomial size blow up. In contrast, we show that interval formulas are different from ROFs and also admit depth reduction which is not known in the case of strict-interval ABPs.Comment: 20 pages, 3 figure