Constructive algorithms and lower bounds for guillotine cuttable orthogonal bin packing problems

The d-dimensional bin packing problem (OBPP-d) is the problem of finding the minimum number of containers needed to contain a set of orthogonally packed d-dimensional rectangular boxes. In OBPP-d solvers two subproblems are crucial: Calculating lower bounds and solving the decision problem (OPP-d) of determining if a set of boxes can be orthogonally packed into a single container. This thesis focuses on these two subproblems with an extra requirement attached: All packings must be guillotine cuttable. That is, the containers must be able to be split into n pieces, each holding a box, by recursively cutting them with face orthogonal cuts. We present an extension of a framework by Fekete and Schepers for the guillotine cutting requirement, prove that the decision problem is NP-hard and prove a worst-case performance for the corresponding OBPP-2. A new type of packing property, sticky cutting, is presented and an algorithm for sticky cuttings based on the framework by Fekete and Schepers is described. Inspired by the nature of guillotine packings, a new tree representation eliminating redundancy is presented along with a proof-of-concept brute-force algorithm for generating all such trees.