Embedding Divisor and Semi-Prime Testability in f-vectors of polytopes

TL;DR

This paper explores the computational complexity of problems related to f-vectors of polytopes, revealing hardness results in general but polynomial-time solvability for simplicial polytopes under certain conditions.

Contribution

It establishes the hardness of divisor and semi-prime testability problems for f-vectors of polytopes and shows these problems are solvable in polynomial time for simplicial polytopes in high dimensions.

Findings

Hardness results for general polytopes' f-vectors.

Polynomial-time algorithms for simplicial polytopes.

Complexity differences conditioned on prime density and P≠NP.

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 $d$.