Lower Bounds for Set-Multilinear Branching Programs
CoRR(2023)
摘要
In this paper, we prove the first super-polynomial and, in fact,
exponential lower bound for the model of sum of ordered
set-multilinear algebraic branching programs, each with a possibly different
ordering (∑𝗌𝗆𝖠𝖡𝖯). In particular, we give an explicit polynomial
such that any ∑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/ loglog 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 ω(log n). Prior to our work, showing such lower
bounds was open irrespective of the assumption on the degree.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要