Lower Bounds for Special Cases of Syntactic Multilinear ABPs

摘要

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.