Lower Bounds for Set-Multilinear Branching Programs
Prerona Chatterjee, Deepanshu Kush, Shubhangi Saraf, Amir Shpilka
TL;DR
This work advances algebraic complexity by proving super-polynomial lower bounds for the model $\sum\mathsf{smABP}$ on explicit set-multilinear polynomials of degree $d$, with $nd$ variables and $d=\omega(\log n)$, and exponential bounds when $d=\poly(n)$. The authors develop a refined partial-derivative framework based on relative rank and arc-partitions, enabling hardness escalation from restricted models to general ABPs and connecting to the Valiant framework via $F_{n,d}$ and $G_{n,d}$ polynomials. They establish tightness results for ABP set-multilinearization and demonstrate near-optimal separations between ordered and unordered smABPs, enhancing our understanding of syntactic restrictions and their impact on lower bounds. The results strengthen the program of deriving super-polynomial lower bounds for ABPs through low-degree hardness (as in BDS23) and provide new tools (arc-partitions, many-violations) that may influence future progress toward resolving Valiant’s conjecture on the permanent.
Abstract
In this paper, we prove super-polynomial lower bounds for the model of \emph{sum of ordered set-multilinear algebraic branching programs}, each with a possibly different ordering ($\sum \mathsf{smABP}$). Specifically, we give an explicit $nd$-variate polynomial of degree $d$ such that any $\sum \mathsf{smABP}$ computing it must have size $n^{ω(1)}$ for $d$ as low as $ω(\log n)$. Notably, this constitutes the first such lower bound in the low degree regime. Moreover, for $d = \poly(n)$, we demonstrate an exponential lower bound. 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 (to appear in TAMC, 2024), 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. Our results strengthen the works of Arvind \& Raja (Chic. J. Theor. Comput. Sci., 2016) and Bhargav, Dwivedi \& Saxena (to appear in TAMC, 2024), as well as the works of Ramya \& Rao (Theor. Comput. Sci., 2020) and Ghoshal \& Rao (International Computer Science Symposium in Russia, 2021), each of which established lower bounds for related or restricted versions of this model. They also strongly answer a question from the former two, which asked to prove super-polynomial lower bounds for general $\sum \mathsf{smABP}$.