Unexplained variability in computational cost of flip graph search paths

Ascertain the reasons why certain flip graph search runs—corresponding to specific transitions between matrix multiplication formats constructed from Moosbauer–Poole base schemes—are computationally more expensive than others, and characterize the algorithmic or structural factors that drive this variability.

Background

The authors use flip graph search starting from strong base schemes for formats (5,5,5) and (6,6,6) to explore nearby rectangular formats via extensions and restrictions represented as arrows in Figure 1. They observe substantial differences in computational effort across these paths.

Despite similar goals and comparable dimensions, some transitions consume significantly more compute, and the cause of this variability is not identified in the paper. Clarifying these reasons could inform more efficient search strategies and improved path selection in future work.

References

Some arrows are computationally more expensive than others, for reasons that are unclear.

— Consequences of the Moosbauer-Poole Algorithms  (2505.05896 - Kauers et al., 9 May 2025) in Paragraph following Figure 1 discussing computation time per arrow

Developing bounds that are simultaneously tight, inexpensive to compute, and sensitive to the permitted boundary relationships remains an active area of research. The framework above identifies the condition required for validity and indicates how such bounds enter the exact recursion; designing the most effective relaxation for particular model classes is left for future work.

— GRACE: A General, Exact Changepoint Calculus  (2609.28831 - Grantham, 23 Sep 2026) in Section 6.2, subsection “Lower bounds on the remaining cost”