On the Number of Integer Points in Translated and Expanded Polyhedra

Abstract

We prove that the problem of minimizing the number of integer points in parallel translations of a rational convex polytope in R 6 is NP-hard. We apply this result to show that given a rational convex polytope P ⊂ R 6 , finding the largest integer t s.t. the expansion tP contains fewer than k integer points is also NP-hard. We conclude that the Ehrhart quasi-polynomials of rational polytopes can have arbitrary fluctuations. Integer Point Minimization (IPM) Input:AParametric polytopes P b := {x ∈ R n : Ax ≤ b} were introduced by Kannan [Kan90], who gave a polynomial time algorithm for IPM with k = 0 and n bounded. For larger fixed values k, Aliev, De Loera and Louveaux [ADL16] proved that IPM is also polynomial time by employing the short generating functions technique by Barvinok and Woods [BW03] (see also [Bar06b,Bar08]). The following problem is an especially attractive special case:

The Generalized Frobenius Problem via Restricted Partition Functions

“…, a n and k are the input? Nguyen and Pak [14] prove that this is NP-hard in the general setting of f (t) = |tP ∩ Z n |, disproving a conjecture from [1]. However, to do this, they construct a polytope P ⊆ R 6 whose f (t) varies wildly across the constituent polynomials, which is not true for our Frobenius f (t) (see Theorem 8).…”