A Truly Subcubic Combinatorial Algorithm for Induced 4-Cycle Detection
Amir Abboud, Shyan Akmal, Nick Fischer
TL;DR
We address the problem of detecting induced $C_4$ cycles in $n$-node graphs. The authors introduce a novel, deterministic, combinatorial approach based on a clique-decomposition into large clusters plus a sparse remainder, followed by win/win subroutines that detect induced $C_4$ across cluster configurations (two, three, and four clusters) and within the remainder. The main result is a truly subcubic algorithm with runtime $ ilde{O}(n^{3-1/6})$, which separates induced $C_4$ detection from triangle detection under the BMM hypothesis and yields the first nontrivial deterministic subcubic algorithm for this problem. The technique leverages ordered-cluster structures, orthogonal range queries, and a layered cluster decomposition, connecting to Erdős–Hajnal considerations and suggesting new directions for induced subgraph detection via clique- and independent-set decompositions. This work thus advances understanding of induced pattern detection and provides a framework that could influence both theory and practical graph analysis.
Abstract
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.