Listing Even Cycles Faster than the Submodular-Width Barrier

Abstract

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 $k\ge 3$. 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.