We study a natural geometric variant of the classic Knapsack problem called 2D-Knapsack: we are given a set of axis-parallel rectangles and a rectangular bounding box, and the goal is to pack as many of these rectangles inside the box without overlap. Naturally, this problem is NP-complete. Recently, Grandoni et al. [ESA'19] showed that it is also W[1]-hard when parameterized by the size $k$ of the sought packing, and they presented a parameterized approximation scheme (PAS) for the variant where we are allowed to rotate the rectangles by 90{\textdegree} before packing them into the box. Obtaining a PAS for the original 2D-Knapsack problem, without rotation, appears to be a challenging open question. In this work, we make progress towards this goal by showing a PAS under the following assumptions: - both the box and all the input rectangles have integral, polynomially bounded sidelengths; - every input rectangle is wide -- its width is greater than its height; and - the aspect ratio of the box is bounded by a constant.Our approximation scheme relies on a mix of various parameterized and approximation techniques, including color coding, rounding, and searching for a structured near-optimum packing using dynamic programming.

翻译：我们研究经典背包问题的一个自然几何变体，称为二维背包问题：给定一组轴平行矩形和一个矩形边界框，目标是在不重叠的情况下将尽可能多的矩形放入该框内。显然，该问题是NP完全的。最近，Grandoni等人[ESA'19]表明，当以所求装填大小$k$为参数时，该问题也是W[1]-难的，并针对允许在装填前将矩形旋转90{\textdegree}的变体提出了一个参数化近似方案(PAS)。对于无旋转的原始二维背包问题，获得PAS似乎是一个具有挑战性的开放问题。在这项工作中，我们通过展示在以下假设下存在PAS，向此目标取得了进展：- 框和所有输入矩形的边长均为整数且受多项式有界；- 每个输入矩形都是宽的——其宽度大于高度；- 框的宽高比受常数限制。我们的近似方案依赖于多种参数化和近似技术的混合，包括色彩编码、舍入以及使用动态规划搜索结构化的近最优装填。