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arXiv · 2311.07214

Forall-exist statements in pseudopolynomial time

Eleonore Bach·Friedrich Eisenbrand·Thomas Rothvoss·Robert Weismantel
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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. $Wx \leq b$. 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 $(m \Delta)^{O(m^2)}$, where $\Delta = \|W\|_\infty$. Furthermore it applies to general convex sets $Q$.

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Eleonore Bach, Friedrich Eisenbrand, Thomas Rothvoss, Robert Weismantel. 2023-11-13. Forall-exist statements in pseudopolynomial time. https://arxiv.org/abs/2311.07214

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