Approximation algorithms for orthogonal packing problems for hypercubes

Orthogonal packing problems are natural multidimensional generalizations of the classical bin packing problem and knapsack problem and occur in many different settings. The input consists of a set I={r1,…,rn} of d-dimensional rectangular items ri=(ai,1,…,ai,d) and a space Q. The task is to pack the items in an orthogonal and non-overlapping manner without using rotations into the given space. In the strip packing setting the space Q is given by a strip of bounded basis and unlimited height. The objective is to pack all items into a strip of minimal height. In the knapsack packing setting the given space Q is a single, usually unit sized bin and the items have associated profits pi. The goal is to maximize the profit of a selection of items that can be packed into the bin.We mainly focus on orthogonal knapsack packing restricted to hypercubes and our main results are a (5/4+ϵ)-approximation algorithm for two-dimensional hypercube knapsack packing, also known as square packing, and a (1+1/2d+ϵ)-approximation algorithm for d-dimensional hypercube knapsack packing. In addition we consider d-dimensional hypercube strip packing in the case of a bounded ratio between the shortest and longest side of the basis of the strip. We derive an asymptotic polynomial time approximation scheme (APTAS) for this problem. Finally, we present an algorithm that packs hypercubes with a total profit of at least into a large bin (the size of the bin depends on ϵ). This problem is known as hypercube knapsack packing with large resources. A preliminary version was published in Harren [Rolf Harren, Approximating the orthogonal knapsack problem for hypercubes, in: ICALP: Proc. 33rd International Colloquium on Automata, Languages and Programming, 2006, pp. 238–249] but especially for the latter two approximation schemes no details were given due to page limitations.