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.
翻译：在本说明中,我们提出了一个算法,将所有4美元的周期都列在一个以美元计时的图表中,$$(n)2,m ⁇ 4/3}+t(美元)美元是美元的数字。值得注意的是,这将四美元的周期列表与三角列表分开,因为三角列表的金额(n)3,m ⁇ 3/2}+)+1-o(1)}(美元)在3美元-SUM的假设下较低约束值之下。我们的上界条件很紧,因为(1)一美元(n)2,m ⁇ 4/3})是已知最能探测到的约4美元的周期,(2)它与最近的(n)美元(n)3,m ⁇ 3/2}+t)1-o(1)美元相匹配,因为三角列表的金额(n)为3美元(n),m ⁇ 3,m ⁇ 3/2}+_1-o(1)},因为后者的下界最近由Abboud、Bringmann和Fischer[arXiv,20222]证明,并且由Jin和Xu[arXiv,2022]独立确定。在一项独立工作中,Jin和Xu[ARXiv,2022]中,目前也受约束的算。
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