Embedding Stacked Polytopes on a Polynomial-Size Grid

Abstract

We show how to realize a stacked 3D polytope (formed by repeatedly stacking a tetrahedron onto a triangular face) by a strictly convex embedding with its n vertices on an integer grid of size O(n4) x O(n4) x O(n18). We use a perturbation technique to construct an integral 2D embedding that lifts to a small 3D polytope, all in linear time. This result solves a question posed by G unter M. Ziegler, and is the rst nontrivial subexponential upper bound on the long-standing open question of the grid size necessary to embed arbitrary convex polyhedra, that is, about effcient versions of Steinitz's 1916 theorem. An immediate consequence of our result is that O(log n)-bit coordinates suffice for a greedy routing strategy in planar 3-trees.