We study the two-dimensional geometric knapsack problem for convex polygons. Given a set of weighted convex polygons and a square knapsack, the goal is to select the most profitable subset of the given polygons that fits non-overlappingly into the knapsack. We allow to rotate the polygons by arbitrary angles. We present a quasi-polynomial time O(1)-approximation algorithm for the general case and a pseudopolynomial time O(1)-approximation algorithm if all input polygons are triangles, both assuming polynomially bounded integral input data. Additionally, we give a quasi-polynomial time algorithm that computes a solution of optimal weight under resource augmentation—that is, we allow to increase the size of the knapsack by a factor of 1+δ for some δ > 0 but compare ourselves with the optimal solution for the original knapsack. To the best of our knowledge, these are the first results for two-dimensional geometric knapsack in which the input objects are more general than axis-parallel rectangles or circles and in which the input polygons can be rotated by arbitrary angles.

我们研究凸多边形的二维几何背包问题。给定一组加权凸多边形和一个方形背包，目标是选择给定多边形中最有利可图的子集，该子集不重叠地装入背包中。我们允许将多边形旋转任意角度。我们提出了一般情况下的拟多项式时间O (1) 近似算法和所有输入多边形均为三角形时的伪多项式时间O (1) 近似算法，两者都假设多项式有界积分输入数据。此外，我们给出了一种拟多项式时间算法，用于计算资源增加下的最佳权重解，也就是说，对于某些 δ > 0，我们允许将背包的大小增加 1+δ 倍，但将我们自己与原始背包的最优解。据我们所知，这是二维几何背包的第一个结果，其中输入对象比轴平行的矩形或圆形更通用，并且输入多边形可以旋转任意角度。