Go backward to 14 Face Lattice of Combinatorial Polytopes
Go up to 3 Combinatorial Structure
Go forward to 16 Reconstruction of Simple Polytopes

15 f-Vector of Combinatorial Polytopes

Input: Vertex-facet incidence matrix of a polytope P
Output: f-vector of P
Status (general): Open
Status (fixed dim.): Polynomial time
By the remarks on Problem 14, it is clear that the first k entries of the f-vector can be computed in polynomial time for every fixed k.

If the polytope is simplicial and a shelling (or a partition) of its boundary complex is available (see Problems 18 and 19), then one can compute the entire f-vector in polynomial time [66, Chap. 8].