Hardness construction for grounded path homology

Construct a reduction establishing average-case hardness for grounded path homology and related directed persistence theories comparable to the reduction available for directed flag complexes.

Background

The paper notes that orienting an undirected graph according to a total vertex order immediately transfers the clique-complex result to directed flag homology. However, that observation does not apply to grounded path homology or related directed persistence theories. The unresolved issue is whether a comparable construction can be developed for those settings.

References

It remains open whether a comparable construction exists for grounded path homology and related directed persistence theories.

— Average-case hardness of Betti number estimation  (2609.12777 - Strelchuk et al., 11 Sep 2026) in Section 7, subsection “Extensions of the average-case hardness framework,” item “Directed, path, and higher-order complexes”