A New Algorithm for Computing Integer Hulls of 2D Polyhedral Sets

Abstract: The $\texttt{IntegerHull}$ function is part of Maple's $\texttt{PolyhedralSets}$ library, which calculates the integer hull of a given polyhedral set. This algorithm works by translating the supporting hyperplanes of the facets of the input polyhedral set inwards till each hyperplane encounters at least one integer point. The polyhedral set is then divided into smaller regions and a brute force method is applied to find the remaining vertices. There are certain edge case scenarios where the computational cost of the existing algorithm can be high, for which we propose a new algorithm. We translate the supporting hyperplanes of the facets inwards from the (relative) opposite vertex till the hyperplanes encounter at least one integer point. Then, we can follow the same procedure as the old algorithm or use a recursive technique on the smaller regions. The edge case scenarios mentioned above occurs when there are integer points present on the supporting hyperplanes of the facets of the polyhedral set. This increases the region on which the brute force method is applied to find the remaining vertices.