Models and Bounds for Two-Dimensional Level Packing Problems

Models and Bounds for Two-Dimensional Level Packing Problems

We consider two-dimensional bin packing and strip packing problems where the items have to be packed by levels. We introduce new mathematical models involving a polynomial number of variables and constraints, and show that their LP relaxations dominate the standard area relaxations. We then propose new (combinatorial) bounds that can be computed in O(n log n) time. We show that they dominate th...

Approximation algorithms for scheduling and two-dimensional packing problems

This dissertation thesis is concerned with two topics of combinatorial optimization: scheduling and geometrical packing problems. Scheduling deals with the assignment of jobs to machines in a ‘good’ way, for suitable notions of good. Two particular problems are studied in depth: on the one hand, we consider the impact of machine failure on online scheduling, i.e. what are the consequences of th...

Algorithms for Two-Dimensional Bin Packing and Assignment Problems

Models and algorithms for three-stage two-dimensional bin packing

We consider the three-stage two-dimensional bin packing problem (2BP) which occurs in real-world applications such as glass, paper, or steel cutting. We present new integer linear programming formulations: Models for a restricted version and the original version of the problem are developed. Both involve polynomial numbers of variables and constraints only and effectively avoid symmetries. Thos...

Improved Absolute Approximation Ratios for Two-Dimensional Packing Problems

We consider the two-dimensional bin packing and strip packing problem, where a list of rectangles has to be packed into a minimal number of rectangular bins or a strip of minimal height, respectively. All packings have to be non-overlapping and orthogonal, i.e., axisparallel. Our algorithm for strip packing has an absolute approximation ratio of 1.9396 and is the first algorithm to break the ap...