Forall-exist statements in pseudopolynomial time

Given a convex set $Q \subseteq R^m$ and an integer matrix $W \in Z^{m \times n}$, we consider statements of the form $ \forall b \in Q \cap Z^m$ $\exists x \in Z^n$ s.t. $Wx \leq b$. Such statements can be verified in polynomial time with the algorithm of Kannan and its improvements if $n$ is fixed and $Q$ is a polyhedron. The running time of the best-known algorithms is doubly exponential in~$n$. In this paper, we provide a pseudopolynomial-time algorithm if $m$ is fixed. Its running time is $(m \Delta)^{O(m^3)}$, where $\Delta = \|W\|_\infty$. Furthermore it applies to general convex sets $Q$. Second, we provide new upper bounds on the \emph{diagonal} as well as the \emph{polyhedral Frobenius} number, two recently studied forall-exist problems.