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 et al. (2001) so as to achieve this ratio. (II) We prove that any collection R of n disjoint1 axis-parallel rectangles drawn on a planar piece of material Q is cu...

This paper proposes a cake-cutting protocol using cryptography when the cake is a heterogeneous good that is represented by an interval on a real line. Although the Dubins-Spanier moving-knife protocol with one knife achieves simple fairness, all players must execute the protocol synchronously. Thus, the protocol cannot be executed on asynchronous networks such as the Internet. We show that the...

We extend the classic cake-cutting problem to a situation in which the ”cake” is divided among families. Each piece of cake is owned and used simultaneously by all members of the family. A typical example of such a cake is land. We examine three ways to assess the fairness of such a division, based on the classic no-envy criterion: (a) Average envy-freeness means that for each family, the avera...

An unceasing problem of our prevailing society is the fair division of goods. The problem of fair cake cutting is dividing a heterogeneous and divisible resource, the cake, among n players who value pieces according to their own measure function. The goal is to assign each player a not necessarily connected part of the cake that the player evaluates at least as much as her proportional share. I...

We describe a new hierarchical 2D-guillotine Cutting Stock Problem. In contrast to the classic cutting stock problem, waste is not an issue. The problem relates to the removal of a defective part and assembly of the remaining parts into homogeneous size blocks. The context is the packing stages of cake manufacturing. The company’s primary objective is to minimise total processing time at the su...

We consider a Two-Dimensional Cutting Stock Problem where stock of different sizes is available, and a set of rectangular items has to be obtained through two-staged guillotine cuts. We propose a heuristic algorithm, based on column generation, which requires as subproblem the solution of a Two-Dimensional Knapsack Problem with two-staged guillotines cuts. A further contribution of the paper co...

In the cutting stock problem (CSP) a given order for smaller pieces has to be cut from larger stock material in such a way that the number of stock material needed is minimal. Based on the classical integer linear programming model the common solution technique consists of solving the corresponding continuous relaxation problem followed by several heuristics which construct integer solutions. I...