In this paper we investigate the following problem: Given two convex shapes Pin and Pout, where Pin is completely contained in Pout, we wish to find a sequence of ‘guillotine cuts’ to cut out Pin from Pout such that the total length of the cutting sequence is minimized. This problem has applications in stock cutting where a particular shape or design (in this case the polygon Pin) needs to be cut out of a given piece of parent material (the polygon Pout) using only guillotine cuts and where it is desired to minimize the cutting sequence length to improve the cutting time required per piece. We first prove some properties of the optimal solution to the problem and then give an approximation scheme for the problem that, given an error range δ, produces a cutting sequence whose total length is at most δ more than that of the optimal cutting sequence. Then it is shown that this problem has optimal solutions that lie in the algebraic extension of the field that the input data belong to – hence due to this algebraic nature of the problem, an approximation scheme is the best that can be achieved. Extensions of these results are also studied in the case where the polygons Pin and Pout are non-convex.
