A Parameterized Approximation Scheme for the Geometric Knapsack Problem with Wide Items

Abstract

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° 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.