Lower bounds for special cases of syntactic multilinear ABPs

Abstract

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 source to the sink node. Proving super-polynomial lower bounds for syntactic multilinear ABPs remains a challenging open question in Algebraic Complexity Theory. The current best lower bound for smABPs is only quadratic in the number of variables [1].In this article, we develop a new approach for proving syntactic multilinear branching program lower bounds: Convert the smABP into an equivalent multilinear formula with a super-polynomial blow-up in size and then exploit the structural limitations of the resulting formula to obtain an upper bound on the rank of partial derivative matrix of the polynomial computed by the smABP.Using this approach, we prove exponential lower bounds for special cases of smABPs namely sum of Read-Once Oblivious smABPs, multilinear r-pass ABPs and α-set-multilinear ABPs. En route, we also prove an exponential lower bound for a special class of syntactic multilinear arithmetic circuits using a similar approach.