Carving Polytopes with Saws in 3D

Abstract

We investigate the problem of carving an $n$-face triangulated three-dimensional polytope using a tool to make cuts modelled by either a half-plane or sweeps from an infinite ray. In the case of half-planes cuts, we present a deterministic algorithm running in $O(n^2)$ time and a randomized algorithm running in $O(n^{3/2+\varepsilon})$ expected time for any $\varepsilon>0$. In the case of cuts defined by sweeps of infinite rays, we present an algorithm running in $O(n^5)$ time.