We present the first truly subcubic, combinatorial algorithm for detecting an induced $4$-cycle in a graph. The running time is $O(n^{2.84})$ on $n$-node graphs, thus separating the task of detecting induced $4$-cycles from detecting triangles, which requires $n^{3-o(1)}$ time combinatorially under the popular BMM hypothesis. Significant work has gone into characterizing the exact time complexity of induced $H$-detection, relative to the complexity of detecting cliques of various sizes. Prior work identified the question of whether induced $4$-cycle detection is triangle-hard as the only remaining case towards completing the lowest level of the classification, dubbing it a "curious" case [Dalirrooyfard, Vassilevska W., FOCS 2022]. Our result can be seen as a negative resolution of this question. Our algorithm deviates from previous techniques in the large body of subgraph detection algorithms and employs the trendy topic of graph decomposition that has hitherto been restricted to more global problems (as in the use of expander decompositions for flow problems) or to shaving subpolynomial factors (as in the application of graph regularity lemmas). While our algorithm is slower than the (non-combinatorial) state-of-the-art $\tilde{O}(n^ω)$-time algorithm based on polynomial identity testing [Vassilevska W., Wang, Williams, Yu, SODA 2014], combinatorial advancements often come with other benefits. In particular, we give the first nontrivial deterministic algorithm for detecting induced $4$-cycles.

翻译：我们提出了一种真正亚三次的组合算法，用于检测图中的诱导四环。该算法在具有n个节点的图上运行时间为O(n^{2.84})，从而将检测诱导四环的任务与检测三角形区分开来——根据流行的BMM假设，后者在组合意义上需要n^{3-o(1)}的时间。已有大量研究致力于刻画检测诱导子图H的精确时间复杂度，特别是相对于检测不同规模团图的复杂度。先前工作将“诱导四环检测是否与三角形检测同等困难”这一问题视为完成最低层次分类的唯一遗留案例，并将其称为一个“奇特”案例[Dalirrooyfard, Vassilevska W., FOCS 2022]。我们的结果可视为对该问题的否定性解答。我们的算法区别于大量子图检测算法中的既有技术，采用了当前热门的图分解方法——该方法此前主要局限于更全局性的问题（如利用扩展图分解处理流问题）或用于削减亚多项式因子（如图正则引理的应用）。虽然我们的算法速度慢于基于多项式恒等式测试的非组合最优算法（时间复杂度为Õ(n^ω)）[Vassilevska W., Wang, Williams, Yu, SODA 2014]，但组合算法的进展往往伴随其他优势。特别地，我们给出了首个非平凡的确定性算法用于检测诱导四环。