Approximation Algorithms for Rectangle Packing Problems (PhD Thesis)

Abstract

In rectangle packing problems we are given the task of placing axis-aligned rectangles in a given plane region, so that they do not overlap with each other. In Maximum Weight Independent Set of Rectangles (MWISR), their position is given and we can only select which rectangles to choose, while trying to maximize their total weight. In Strip Packing (SP), we have to pack all the given rectangles in a rectangular region of fixed width, while minimizing its height. In 2-Dimensional Geometric Knapsack (2DGK), the target region is a square of a given size, and our goal is to select and pack a subset of the given rectangles of maximum weight. We study a generalization of MWISR and use it to improve the approximation for a resource allocation problem called bagUFP. We revisit some classical results on SP and 2DGK, by proposing a framework based on smaller containers that are packed with simpler rules; while variations of this scheme are indeed a standard technique in this area, we abstract away some of the problem-specific differences, obtaining simpler algorithms that work for different problems. We obtain improved approximations for SP in pseudo-polynomial time, and for a variant of 2DGK where one can to rotate the rectangles by 90°. For the latter, we propose the first algorithms with approximation factor better than 2. For the main variant of 2DGK (without rotations), a container-based approach seems to face a natural barrier of 2 in the approximation factor. Thus, we consider a generalized kind of packing that combines container packings with another packing problem that we call L-packing problem, where we have to pack rectangles in an L-shaped region of the plane. By finding a (1 + ε)-approximation for this problem and exploiting the combinatorial structure of 2DGK, we obtain the first algorithms that break the barrier of 2 for the approximation factor of this problem.