Lower Bounds for Set-Multilinear Branching Programs

In this paper, we prove the first \emph{super-polynomial} and, in fact, \emph{exponential} lower bound for the model of \emph{sum of ordered set-multilinear algebraic branching programs}, each with a possibly different ordering ($\sum \mathsf{smABP}$). In particular, we give an explicit polynomial such that any $\sum \mathsf{smABP}$ computing it must have exponential size. This result generalizes the seminal work of Nisan (STOC 1991), which proved an exponential lower bound for a single ordered set-multilinear ABP. The significance of our lower bounds is underscored by the recent work of Bhargav, Dwivedi, and Saxena (2023), which showed that super-polynomial lower bounds against a sum of ordered set-multilinear branching programs -- for a polynomial of sufficiently low degree -- would imply super-polynomial lower bounds against general ABPs, thereby resolving Valiant's longstanding conjecture that the permanent polynomial can not be computed efficiently by ABPs. More precisely, their work shows that if one could obtain such lower bounds when the degree is bounded by $O(\log n/ \log \log n)$, then it would imply super-polynomial lower bounds against general ABPs. In this paper, we show super-polynomial lower bounds against this model for a polynomial whose degree is as small as $\omega(\log n)$. Prior to our work, showing such lower bounds was open \emph{irrespective} of the assumption on the degree.