A parameterized approximation scheme for the 2D-Knapsack problem with wide items

TL;DR

This paper develops a parameterized approximation scheme for the 2D-Knapsack problem with wide rectangles, addressing a challenging open problem by combining advanced techniques under specific assumptions.

Contribution

It introduces a PAS for the 2D-Knapsack problem without rotation, assuming polynomially bounded sizes, wide rectangles, and a bounded aspect ratio.

Findings

Provides a PAS under specific geometric and size assumptions.

Uses a combination of color coding, rounding, and dynamic programming techniques.

Progresses towards solving the open problem of approximating 2D-Knapsack without rotation.

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{\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.