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 regime where we prove this computational difference (conditioned on standard conjectures on the density of primes and on $P\neq NP$) is when the dimension $d$ tends to infinity and the number of facets is linear in $d$.