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.

翻译：本文证明了有序集合多重线性代数分支程序（每个程序可能具有不同顺序，即∑ smABP）模型的第一个超多项式（实际上是指数）下界。具体而言，我们给出了一个显式多项式，使得任何计算该多项式的∑ smABP都必须具有指数规模。这一结果推广了Nisan（STOC 1991）的开创性工作，后者证明了单个有序集合多重线性ABP的指数下界。我们下界的重要性因Bhargav、Dwivedi和Saxena（2023）的近期工作而更加凸显：他们证明了针对足够低度多项式之和的有序集合多重线性分支程序的超多项式下界，将蕴含一般ABP的超多项式下界，从而解决Valiant长期存在的猜想——永久多项式无法被ABP高效计算。更精确地说，其工作表明：若能获得度受限于O(log n / log log n)时的此类下界，则将推导出一般ABP的超多项式下界。本文针对次数低至ω(log n)的多项式证明了该模型的超多项式下界。在本文之前，无论对次数做何假设，此类下界的证明均是未解决的公开问题。