A Truly Subcubic Combinatorial Algorithm for Induced 4-Cycle Detection

TL;DR

This paper introduces the first truly subcubic combinatorial algorithm for detecting induced 4-cycles in graphs, achieving a running time of O(n^{2.84}) and providing a deterministic alternative to algebraic methods.

Contribution

It presents the first truly subcubic combinatorial algorithm for induced 4-cycle detection, separating it from triangle detection complexity and offering a deterministic approach.

Findings

Achieves O(n^{2.84}) running time for induced 4-cycle detection.

Separates induced 4-cycle detection complexity from triangle detection.

Provides the first deterministic algorithm for this problem.

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 $\tilde{O}(n^{\omega})$-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.