A Truly Subcubic Combinatorial Algorithm for Induced 4-Cycle Detection

2025-07-24

AI Summary

This paper investigates the computational complexity of induced $C_4$ detection, addressing the long-standing conjecture that it is “triangle-hard”—i.e., asymptotically as hard as triangle detection. Method: The authors devise the first truly subcubic, deterministic, purely combinatorial algorithm for induced $C_4$ detection on $n$-vertex graphs, running in $O(n^{2.84})$ time—breaking the conventional $Omega(n^3)$ combinatorial lower bound. The algorithm leverages a novel graph decomposition framework that integrates expander decomposition with fine-grained local structural analysis, avoiding randomization or algebraic techniques. Contribution/Results: The result refutes the conjecture that induced $C_4$ detection is triangle-hard, resolving its fine-grained complexity classification. Moreover, it extends the graph decomposition paradigm beyond global graph properties to local subgraph detection, yielding a new combinatorial tool for subgraph discovery in sparse graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $ ilde{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.

Problem

• First subcubic combinatorial algorithm for induced 4-cycle detection
• Resolves whether induced 4-cycle detection is triangle-hard
• Provides first nontrivial deterministic algorithm for this task

Innovation

• First truly subcubic combinatorial algorithm
• Uses graph decomposition techniques innovatively
• Provides first nontrivial deterministic solution