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 $n^{\omega(1)}$ for $d$ as low as $\omega(\log n)$. 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 $\sum \mathsf{smABP}$.

翻译：本文证明了有序集合多线性代数分支程序之和模型（每个程序可能具有不同的排序，记为$\sum \mathsf{smABP}$）的超多项式下界。具体而言，我们构造了一个显式的$nd$元$d$次多项式，使得对于低至$\omega(\log n)$的$d$，任何计算该多项式的$\sum \mathsf{smABP}$都必须具有规模$n^{\omega(1)}$。值得注意的是，这是低度机制下的首个此类下界。此外，对于$d = \poly(n)$的情况，我们证明了一个指数下界。这一结果推广了Nisan（STOC，1991）的开创性工作，该工作证明了单个有序集合多线性ABP的指数下界。我们下界的重要意义在于：Bhargav、Dwivedi和Saxena（即将发表于TAMC，2024）的最新工作表明，对于足够低次的多项式，如果能够证明有序集合多线性分支程序之和模型的超多项式下界，那么这将蕴含一般ABP的超多项式下界，从而解决Valiant关于永久多项式无法被ABP有效计算的长期猜想。更精确地说，他们的工作表明：若能在次数受限于$O(\log n/ \log \log n)$时获得此类下界，则将推出一般ABP的超多项式下界。我们的结果强化了Arvind & Raja（Chic. J. Theor. Comput. Sci.，2016）以及Bhargav、Dwivedi & Saxena（即将发表于TAMC，2024）的工作，同时也加强了Ramya & Rao（Theor. Comput. Sci.，2020）以及Ghoshal & Rao（俄罗斯国际计算机科学研讨会，2021）的工作——这些工作均针对该模型的关联版本或受限版本建立了下界。此外，我们的结果有力地回答了前两篇工作中的问题，该问题要求证明一般$\sum \mathsf{smABP}$的超多项式下界。