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}$I=\{r_1,\dots ,r_n\}$ of d$d$-dimensional rectangular items _ri=(_ai,1,…,_ai,d)$r_i=(a_{i,1},\dots ,a_{i,d})$ and a space Q$Q$. 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$Q$ 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$Q$ is a single, usually unit sized bin and the items have associated profits _pi$p_i$. The goal is to maximize the profit of a selection of items that can be packed into the bin.