Triangle Detection in H-Free Graphs
Amir Abboud, Ron Safier, Nathan Wallheimer

TL;DR
This paper studies combinatorial algorithms for detecting triangles in H-free graphs, classifying which patterns allow subcubic algorithms and developing new methods for embeddable patterns, with specialized algorithms for odd cycles.
Contribution
It introduces a classification of patterns for triangle detection in H-free graphs, developing subcubic combinatorial algorithms for embeddable patterns and specialized algorithms for odd cycles.
Findings
Subcubic combinatorial algorithms for embeddable patterns of size k
Dichotomy theorem for patterns up to size eight
Efficient algorithms for odd cycle detection
Abstract
We initiate the study of combinatorial algorithms for Triangle Detection in -free graphs. The goal is to decide if a graph that forbids a fixed pattern as a subgraph contains a triangle, using only "combinatorial" methods that notably exclude fast matrix multiplication. Our work aims to classify which patterns admit a subcubic speedup, working towards a dichotomy theorem. On the lower bound side, we show that if is not -colorable or contains more than one triangle, the complexity of the problem remains unchanged, and no combinatorial speedup is likely possible. On the upper bound side, we develop an embedding approach that results in a strongly subcubic, combinatorial algorithm for a rich class of "embeddable" patterns. Specifically, for an embeddable pattern of size , our algorithm runs in time, where hides…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
