Shortest-path complexity on cosmohedra and permuto-associahedra

Determine the computational complexity of computing the shortest combinatorial path (graph distance in the 1-skeleton) between two vertices of the cosmohedron or the permuto-associahedron.

Background

The paper observes that these polyhedra have 1-skeleta containing the flip graph of triangulations of convex polygons as a subgraph, suggesting potential hardness results.

However, it is currently unclear whether shortest paths stay within that subgraph; as a result, the authors explicitly state that the overall shortest-path complexity on these polytopes remains open.