AI 中文总结
本文研究全对三角形检测的黑盒归约局限性，证明R-三角形与全边R-三角形无更紧密黑盒归约，还给出三角形与矩阵问题间新归约及相关条件下界，为细粒度证明技术设障碍。
AI 中文摘要
对于任意三分关系 $R\subseteq \mathbb{Z}^3$，$R$-三角形问题是指，给定一个带权图，判断其是否包含一个权重属于 $R$ 中的三元组的三角形；全边 $R$-三角形问题则要求判断每条边是否包含在这样的三角形中。已知对所有 $R$，$R$-三角形与全边 $R$-三角形是亚立方细粒度等价的[Vassilevska W.-Williams'10]。不过，尽管人们推测这些问题是紧密等价的，但该归约仅表明：若 $R$-三角形存在某个 $\epsilon>0$ 的 $O(n^{3-\epsilon})$ 时间算法，则全边 $R$-三角形存在 $O(n^{3-\epsilon/3})$ 时间算法。本文为紧密等价提供了一个强无条件障碍：[Vassilevska W.-Williams'10]的归约对适用于任意 $R$ 的黑盒归约而言是最优的。我们还给出了多种 $R$-三角形问题之间黑盒归约的进一步结果。我们的正面结果产生了若干类三角形与矩阵问题之间的新归约，例如，我们证明计算相等或优势乘积的 $O(n^{2.53})$ 时间算法，将意味着对已知布尔 $(\min, +)$-乘积算法的改进，这为优势乘积和相等乘积提供了首个条件下界。我们的负面结果可视为对自然细粒度证明技术的障碍。除了 $R$-三角形与全边 $R$-三角形之间无法实现更紧密等价的结果外，我们还表明，尽管人们推测三角形计数与二元整数矩阵乘法、布尔矩阵乘法与列出 $n^2$ 个三角形等所有这些问题都应等价，但没有适当的“黑盒”归约能够证明三角形计数与二元整数矩阵乘法之间的亚立方等价，或布尔矩阵乘法与列出 $n^2$ 个三角形之间的紧密等价，等等。
英文摘要
For any tripartite relation $R\subseteq \mathbb{Z}^3$, the $R$-Triangle problem asks, given an edge-weighted graph, whether it contains a triangle whose weights form a triple in $R$. The All-Edge $R$-Triangle problem asks to determine for every edge whether it is contained in such a triangle. It is known that $R$-Triangle and All-Edge $R$-Triangle are subcubically fine-grained equivalent for every $R$ [Vassilevska W.-Williams'10]. However, while it is conjectured that these problems are tightly equivalent, this reduction only shows that if $R$-Triangle has an $O(n^{3-ε})$-time algorithm for some $ε>0$, then All-Edge $R$-Triangle has an $O(n^{3-ε/3})$-time algorithm. This paper provides a strong unconditional barrier to a tight equivalence: the reduction of [Vassilevska W.-Williams'10] is optimal for black-box reductions that work for arbitrary $R$. We give further results about black-box reductions between a variety of $R$-triangle problems. Our positive results yield new reductions between several classes of triangle and matrix problems --- for instance, we demonstrate that an $O(n^{2.53})$-time algorithm for computing equality or dominance product would imply an improvement on known algorithms for computing boolean $(\min, +)$-product, giving the first conditional lower bound for dominance and equality product. Our negative results can be thought of as barriers against natural fine-grained proof techniques. Besides the result that a tighter equivalence between $R$-Triangle and All-Edge $R$-Triangle is not possible, we also show that no appropriately "black-box" reductions are capable of demonstrating a subcubic equivalence between triangle counting and binary integer matrix multiplication, or a tight equivalence between boolean matrix multiplication and listing $n^2$ triangles, and more, despite the fact that all of these equivalences are conjectured to hold.
Comments32 pages
