Decomposing Polygons Into Diameter Bounded Components
Chris Worman · Canadian Conference on Computational Geometry · 2003
A decomposition of a polygon P is a set of polygons whose geometric union is exactly P . We consider the problem of decomposing a polygon, which may contain holes, using subpolygons that have a bounded diameter. We show that this problem is NP-complete via a reduction from P lanar 3, 4SAT. Polygon decomposition problems arise in applications where objects represented by polygons need to be subdivided for the sake of tractability. Many variations of decomposition problems have recieved attention in the literature. The reader is directed towards [5] for a synopsis of recent polygon decomposition results. Of particular interest are those results concerning the decomposition of non-simple polygons. The problem of minimally decomposing a polygon that may contain holes has proven to be difficult, and is typically NP-hard. Bounding box heuristics are commonly used in object intersection algorithms. It has been shown that these algorithms have better performance guarantees when the bounding boxes have similar sizes [6]. This result motivates Damian-Iordache [3] to explore the idea of restricting the diameter of the components in the decomposition of a polygon. Damian-Iordache is able to develop a polynomial time algorithm for partitioning a simple polygon into the minimum number of components that have a maximum diameter of α. Here α is a fixed real number that is part of the input to the partioning algorithm. The problem of decomposing a polygon, which may have holes, with the minimum number of diameter bounded components is conjectured to be NP-hard [3]. We confirm this conjecture by reducing P lanar 3, 4SAT to the corresponding covering and partitioning decision problems.