单色三角形、三角形列表和APSP
Monochromatic Triangles, Triangle Listing and APSP
摘要：
细粒度复杂度的主要假设之一是$n$ -节点图的所有对最短路径（APSP）需要$n^{3-o(1)}$时间。另一个著名的假设是$n$整数的$3$ SUM问题需要$n^{2-o(1)}$时间。虽然$3$ SUM和APSP之间没有直接的归约，但已知它们是相关的：存在一个问题，$(\min,+)$ -卷积以细粒度的方式归约到两者，以及一个问题精确三角形，两者都是细粒度的归约。在本文中，我们发现这两个问题和其他基本问题之间的关系更多。Pagetrajeccu已经证明，在$3$ SUM假设下，$m$ -边图中的所有边稀疏三角形问题需要$m^{4/3-o(1)}$时间。后一个问题要求确定每个边$e$，是否$e$是在一个三角形。它等价于在$m$ -边图中列出$m$三角形的问题，其中$m=\tilde{O}(n^{1.5})$，并且可以在$O(m^{1.41})$时间内解决[Alon et al. '97]与当前矩阵乘法界限，并且如果$ω=2$，则在$\tilde{O}(m^{4/3})$时间内。我们表明，可以减少精确三角形的所有边稀疏三角形，表明所有边稀疏三角形（因此三角列表）需要$m^{4/3-o(1)}$时间也假设APSP假设。这使我们能够为许多以前在$3$ SUM假设下已知很难的动态问题提供APSP硬度。我们还考虑了以前研究的所有边单色三角形问题。通过工作[林肯等人。'20]，我们对全边稀疏三角形的结果意味着，如果全边单色三角形问题具有$ε>0$的$O(n^{2.5-ε})$时间算法，则APSP和$3$ SUM假设都是假的。我们还将这个问题与其他"中间“问题联系起来，这些问题的运行时间介于$O(n^ω)$和$O(n^3)$之间，例如Max-Min乘积问题。
英文摘要：
One of the main hypotheses in fine-grained complexity is that All-Pairs Shortest Paths (APSP) for $n$-node graphs requires $n^{3-o(1)}$ time. Another famous hypothesis is that the $3$SUM problem for $n$ integers requires $n^{2-o(1)}$ time. Although there are no direct reductions between $3$SUM and APSP, it is known that they are related: there is a problem, $(\min,+)$-convolution that reduces in a fine-grained way to both, and a problem Exact Triangle that both fine-grained reduce to. In this paper we find more relationships between these two problems and other basic problems. Pătraşcu had shown that under the $3$SUM hypothesis the All-Edges Sparse Triangle problem in $m$-edge graphs requires $m^{4/3-o(1)}$ time. The latter problem asks to determine for every edge $e$, whether $e$ is in a triangle. It is equivalent to the problem of listing $m$ triangles in an $m$-edge graph where $m=\tilde{O}(n^{1.5})$, and can be solved in $O(m^{1.41})$ time [Alon et al.'97] with the current matrix multiplication bounds, and in $\tilde{O}(m^{4/3})$ time if $ω=2$. We show that one can reduce Exact Triangle to All-Edges Sparse Triangle, showing that All-Edges Sparse Triangle (and hence Triangle Listing) requires $m^{4/3-o(1)}$ time also assuming the APSP hypothesis. This allows us to provide APSP-hardness for many dynamic problems that were previously known to be hard under the $3$SUM hypothesis. We also consider the previously studied All-Edges Monochromatic Triangle problem. Via work of [Lincoln et al.'20], our result on All-Edges Sparse Triangle implies that if the All-Edges Monochromatic Triangle problem has an $O(n^{2.5-ε})$ time algorithm for $ε>0$, then both the APSP and $3$SUM hypotheses are false. We also connect the problem to other ``intermediate'' problems, whose runtimes are between $O(n^ω)$ and $O(n^3)$, such as the Max-Min product problem.
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