A classic result of Alon, Yuster, and Zwick (AYZ, Algorithmica 1997) shows that all $2k$-cycles in an $m$-edge graph can be listed in $\tilde O(m^{2-1/k}+t)$ time, where $t$ is the output size. This bound underlies the {\em submodular width} of Marx (JACM 2013) and the PANDA framework of Abo Khamis, Ngo, and Suciu (PODS 2017), which extend AYZ to arbitrary conjunctive queries with degree constraints. A central open question is whether combinatorial algorithms can beat the submodular-width barrier. Bringmann and Gorbachev (STOC 2025) gave lower-bound evidence that submodular width may be optimal for general conjunctive queries under combinatorial algorithms. The picture changes for $2k$-cycles on undirected graphs, whose queries have self-joins and symmetric EDBs: recent works improve on AYZ for even-cycle detection and listing. Pinning down the complexity of $C_{2k}$-detection and listing is thus a natural step toward overcoming the submodular-width barrier for such queries. For detection, Dahlgaard, Knudsen, and St{ö}ckel (STOC 2017) solved $C_{2k}$-detection in $\tilde O(m^{2k/(k+1)})$ time. Listing is harder. Jin and Xu (STOC 2023), and independently Abboud, Khoury, Leibowitz, and Safier (FSTTCS 2023), listed 4-cycles in $\tilde O(m^{4/3}+t)$ time; Vassilevska~Williams and Westover (ITCS 2025) listed 6-cycles in $\tilde O(m^{8/5}+t)$ time, improving the AYZ bounds of $\tilde O(m^{3/2})$ and $\tilde O(m^{5/3})$. The general case has remained open for 30 years. Building on these works, we list $2k$-cycles in $\tilde O(m^{(2k^2-k+1)/(k^2+1)}+t)$ time, improving AYZ for every $k\geq 3$. The key ingredient is an \emph{asymmetric supersaturation} result for even cycles. Our algorithms use only join and project operators over multiple tree-decomposition plans, making them naturally implementable in database systems, in contrast to prior BFS-based graph approaches.

翻译：Alon、Yuster和Zwick（AYZ，Algorithmica 1997）的经典结果表明，在具有$m$条边的图中，所有$2k$-环可在$\tilde O(m^{2-1/k}+t)$时间内被列举，其中$t$为输出规模。这一界限构成了Marx（JACM 2013）的{\em 子模宽度}及Abo Khamis、Ngo与Suciu（PODS 2017）的PANDA框架的基础——后者将AYZ方法推广至带度约束的任意合取查询。一个核心开放问题是组合算法能否突破子模宽度壁垒。Bringmann与Gorbachev（STOC 2025）给出了下界证据，表明对于一般合取查询，子模宽度在组合算法框架下可能具有最优性。然而对于无向图上的$2k$-环，情形有所不同：此类查询涉及自连接和对称EDB，近期研究已在偶数环检测与列举上改进了AYZ结果。因此，明确$C_{2k}$检测与列举的复杂性，成为突破此类查询子模宽度壁垒的自然步骤。在检测方面，Dahlgaard、Knudsen与Stöckel（STOC 2017）实现了$\tilde O(m^{2k/(k+1)})$时间内的$C_{2k}$检测。列举则更为困难。Jin与Xu（STOC 2023）以及独立工作的Abboud、Khoury、Leibowitz与Safier（FSTTCS 2023）在$\tilde O(m^{4/3}+t)$时间内列举了4-环；Vassilevska Williams与Westover（ITCS 2025）在$\tilde O(m^{8/5}+t)$时间内列举了6-环，分别改进了AYZ的$\tilde O(m^{3/2})$和$\tilde O(m^{5/3})$界。一般情形在三十年来始终悬而未决。基于上述工作，我们实现了$\tilde O(m^{(2k^2-k+1)/(k^2+1)}+t)$时间内列举$2k$-环，对每个$k\geq 3$均改进了AYZ结果。关键技术是偶数环的{\em 非对称过饱和}结论。与先前基于BFS的图算法不同，我们的算法仅使用多树分解计划上的连接与投影操作，使其天然适用于数据库系统实现。