In a seminal work, Chiba and Nishizeki [SIAM J. Comput. `85] developed subgraph listing algorithms for triangles, 4-cycle and $k$-cliques, where $k \geq 3.$ The runtimes of their algorithms are parameterized by the number of edges $m$ and the arboricity $\alpha$ of a graph. The arboricity $\alpha$ of a graph is the minimum number of spanning forests required to cover it. Their work introduces: * A triangle listing algorithm that runs in $O(m\alpha)$ time. * An output-sensitive 4-Cycle-Listing algorithm that lists all 4-cycles in $O(m\alpha + t)$ time, where $t$ is the number of 4-cycles in the graph. * A k-Clique-Listing algorithm that runs in $O(m\alpha^{k-2})$ time, for $k \geq 4.$ Despite the widespread use of these algorithms in practice, no improvements have been made over them in the past few decades. Therefore, recent work has gone into studying lower bounds for subgraph listing problems. The works of Kopelowitz, Pettie and Porat [SODA `16] and Vassilevska W. and Xu [FOCS `20] showed that the triangle-listing algorithm of Chiba and Nishizeki is optimal under the $\mathsf{3SUM}$ and $\mathsf{APSP}$ hypotheses respectively. However, it remained open whether the remaining algorithms were optimal. In this note, we show that in fact all the above algorithms are optimal under popular hardness conjectures. First, we show that the $\mathsf{4}\text{-}\mathsf{Cycle}\text{-}\mathsf{Listing}$ algorithm is tight under the $\mathsf{3SUM}$ hypothesis following the techniques of Jin and Xu [STOC `23], and Abboud, Bringmann and Fishcher [STOC `23] . Additionally, we show that the $k\text{-}\mathsf{Clique}\text{-}\mathsf{Listing}$ algorithm is essentially tight under the exact $k$-clique hypothesis by following the techniques of Dalirooyfard, Mathialagan, Vassilevska W. and Xu [STOC `24]. These hardness results hold even when the number of 4-cycles or $k$-cliques in the graph is small.

翻译：在一项开创性工作中，Chiba和Nishizeki [SIAM J. Comput. `85] 提出了针对三角形、4-环以及 $k$-团（其中 $k \geq 3$）的子图列举算法。其算法的运行时间由图的边数 $m$ 和树密度 $\alpha$ 参数化。图的树密度 $\alpha$ 是覆盖该图所需的最小生成森林数量。他们的工作引入了：* 一个在 $O(m\alpha)$ 时间内运行的三角形列举算法。* 一个输出敏感的4-环列举算法，能在 $O(m\alpha + t)$ 时间内列出所有4-环，其中 $t$ 是图中4-环的数量。* 一个对于 $k \geq 4$，在 $O(m\alpha^{k-2})$ 时间内运行的k-团列举算法。尽管这些算法在实践中被广泛使用，但在过去几十年里，它们并未得到改进。因此，近期的研究转向探究子图列举问题的下界。Kopelowitz、Pettie和Porat [SODA `16] 以及Vassilevska W.和Xu [FOCS `20] 的工作表明，Chiba和Nishizeki的三角形列举算法分别在 $\mathsf{3SUM}$ 和 $\mathsf{APSP}$ 假设下是最优的。然而，其余算法是否最优则一直悬而未决。在本注记中，我们证明，事实上，上述所有算法在流行的硬度猜想下都是最优的。首先，我们借鉴Jin和Xu [STOC `23] 以及Abboud、Bringmann和Fishcher [STOC `23] 的技术，证明在 $\mathsf{3SUM}$ 假设下，$\mathsf{4}\text{-}\mathsf{Cycle}\text{-}\mathsf{Listing}$ 算法是紧的。此外，我们遵循Dalirooyfard、Mathialagan、Vassilevska W.和Xu [STOC `24] 的技术，证明在精确 $k$-团假设下，$k\text{-}\mathsf{Clique}\text{-}\mathsf{Listing}$ 算法本质上是紧的。这些硬度结果即使在图中4-环或 $k$-团的数量很少时也成立。