We study a generalization of the knapsack problem with geometric and vector constraints. The input is a set of rectangular items, each with an associated profit and $d$ nonnegative weights ($d$-dimensional vector), and a square knapsack. The goal is to find a non-overlapping axis-parallel packing of a subset of items into the given knapsack such that the vector constraints are not violated, i.e., the sum of weights of all the packed items in any of the $d$ dimensions does not exceed one. We consider two variants of the problem: $(i)$ the items are not allowed to be rotated, $(ii)$ items can be rotated by 90 degrees. We give a $(2+\epsilon)$-approximation algorithm for this problem (both versions). In the process, we also study a variant of the maximum generalized assignment problem (Max-GAP), called Vector-Max-GAP, and design a PTAS for it.

中文翻译：

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广义多维背包的近似算法

我们研究了具有几何和矢量约束的背包问题的推广。输入是一组矩形项目，每个项目都有相关的利润和$ d $非负权重（$ d $维向量）和一个方形背包。目的是找到一个子集到给定的背包中的一个非重叠的轴平行包装，这样就不会违反向量约束，即，任何$ d $维度中所有包装物品的权重之和不超过一个。我们考虑问题的两个变体：$（i）$不允许旋转项目，$（ii）$可以旋转90度。对于这个问题，我们给出了一个$（2+ \ epsilon）$近似算法（两个版本）。在此过程中，我们还将研究最大广义分配问题（Max-GAP）的一个变体，称为Vector-Max-GAP，并为此设计一个PTAS。