Forall-exist statements in pseudopolynomial time

Abstract

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. $Wx\le b$. 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 $(m\hat{I}”{)}^{O(m^2)}$, where $\hat{I}”=\Vert W{\Vert }_{\infty }$. Furthermore it applies to general convex sets $Q$.