On collections of polygons cuttable with a segment saw

Abstract

(I) Given a cuttable polygon P drawn on a piece of planar material Q, we show how to cut P out of Q by a (small) segment saw with a total length no more than 2.5 times the optimal. We revise the algorithm of Demaine et al. (2001) so as to achieve this ratio. (II) We prove that any collection R of n disjoint axis-parallel rectangles1 is cuttable by at most 4n rays and present an algorithm that runs in O(n log n) time for computing a suitable cutting sequence. In particular the same result holds for cutting with an arbitrary segment saw (of any length). (III) In contrast, we show that there exist collections of disjoint rectangles (in arbitrary orientations) that are uncuttable by a segment saw. We also present various uncuttable collections of disjoint polygons, including triangles.