Embedding Divisor and Semi-Prime Testability in f-Vectors of Polytopes

Abstract

We obtain computational hardness results for f-vectors of polytopes by exhibiting reductions of the problems DIVISOR and SEMI-PRIME TESTABILITY to problems on f-vectors of polytopes. Further, we show that the corresponding problems for f-vectors of simplicial polytopes are polytime solvable. The situation where we prove this computational difference (conditioned on standard conjectures on the density of primes and on \(P \ne NP \)) is when the dimension d tends to infinity and the number of facets is linear in d.