Cutting out polygon collections with a saw.

(I) Given a segment-cuttable polygon P drawn on a planar piece of material Q, we show how to cut P out of Q by a (short) segment saw with a total length of the cuts no more than 2.5 times the optimal. We revise the algorithm of Demaine etal. (2001) so as to achieve this ratio.(II) We prove that any collection R of n disjoint11For brevity, a collection of disjoint polygons refers to a collection of polygons with pairwise disjoint interiors. axis-parallel rectangles drawn on a planar piece of material Q is cuttable by at most 4n rays and present an algorithm that runs in O(nlogn) time for computing a suitable cutting sequence. In particular, the same result holds for cutting with an arbitrary segment saw (of any length).(III) Given a collection P of segment-cuttable polygons drawn on a planar piece of material such that no two polygons in P touch each other, P is always cuttable by a sufficiently short segment saw. We also show that there exist collections of disjoint polygons that are uncuttable by a segment saw.(IV) Given a collection P of disjoint polygons drawn on a planar piece of material Q, we present a polynomial-time algorithm that computes a suitable cutting sequence to cut the polygons in P out of Q using ray cuts when P is ray-cuttable and otherwise reports P as uncuttable.