论文标题
列出4个周期
Listing 4-Cycles
论文作者
论文摘要
在本说明中，我们提出了一种算法，该算法列出了图表中的所有$ 4 $ - cycles $ \ tilde {o}（\ min（n^2，m^2，m^{4/3}）+t）$，其中$ t $是它们的数字。值得注意的是，这将$ 4 $循环清单与三角上列表分开，因为后者具有$（\ min（n^3，m^{3/2}）+t）+t）+t）^{1-o（1）} $下限$ 3 $ - sum-sum-sum-sum-sum-sumpoxulure。 我们的上限有条件地紧密，因为（1）$ o（n^2，m^{4/3}）$是最著名的检测，如果该图具有任何$ 4 $ -Cycle，并且（2）与最近的$（\ min（n^3，m^3，m^{3/2}}）+T） 后者的下限最近被Abboud，Bringmann和Fischer [Arxiv，2022]证明，并独立于Jin和Xu [Arxiv，2022]。 在独立的工作中，Jin和Xu [Arxiv，2022]还提出了具有相同时间结合的算法。
In this note we present an algorithm that lists all $4$-cycles in a graph in time $\tilde{O}(\min(n^2,m^{4/3})+t)$ where $t$ is their number. Notably, this separates $4$-cycle listing from triangle-listing, since the latter has a $(\min(n^3,m^{3/2})+t)^{1-o(1)}$ lower bound under the $3$-SUM Conjecture. Our upper bound is conditionally tight because (1) $O(n^2,m^{4/3})$ is the best known bound for detecting if the graph has any $4$-cycle, and (2) it matches a recent $(\min(n^3,m^{3/2})+t)^{1-o(1)}$ $3$-SUM lower bound for enumeration algorithms. The latter lower bound was proved very recently by Abboud, Bringmann, and Fischer [arXiv, 2022] and independently by Jin and Xu [arXiv, 2022]. In an independent work, Jin and Xu [arXiv, 2022] also present an algorithm with the same time bound.
