Listing Even Cycles Faster than the Submodular-Width Barrier

2026-05-28

AI Summary

This work addresses the problem of efficiently enumerating all even-length cycles of size $2k$ (for $k \geq 3$) in undirected graphs. The authors propose a combinatorial algorithm based on a multi-tree decomposition scheme that leverages join and projection operations, eschewing traditional BFS traversals to better align with database-oriented implementations. The key innovation lies in the introduction of asymmetric hyper-saturation theory, which enables the algorithm to surpass the classical time complexity lower bound established by Alon, Yuster, and Zwick. The resulting algorithm achieves a running time of $\tilde{O}(m^{(2k^2 - k + 1)/(k^2 + 1)} + t)$, improving upon the best-known bound of $\tilde{O}(m^{2 - 1/k} + t)$ for all $k \geq 3$.

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 $k\geq 3$. 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.

Problem

• even cycles
• submodular width
• cycle listing
• conjunctive queries
• graph algorithms

Innovation

• asymmetric supersaturation
• even-cycle listing
• submodular-width barrier
• tree decomposition
• combinatorial algorithm