Lower Bounds for Special Cases of Syntactic Multilinear ABPs

TL;DR

This paper establishes exponential lower bounds for certain restricted classes of syntactic multilinear algebraic branching programs, advancing understanding of their computational limitations.

Contribution

It introduces a new approach to upper bound the rank of the partial derivative matrix, leading to exponential lower bounds for specific smABP subclasses.

Findings

Proves exponential lower bounds for sum of Oblivious Read-Once ABPs

Establishes lower bounds for r-pass multilinear ABPs

Demonstrates super-polynomial lower bounds for certain syntactic multilinear circuits

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 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.