Improved Pseudo-Polynomial-Time Approximation for Strip Packing

TL;DR

This paper presents a new pseudo-polynomial-time approximation algorithm for the strip packing problem, improving the approximation ratio from 7/5+ε to 4/3+ε, and extends the approach to rotated rectangles.

Contribution

It introduces a novel repacking technique that enhances approximation ratios for strip packing in pseudo-polynomial time, surpassing previous barriers.

Findings

Achieves a (4/3+ε)-approximation algorithm for strip packing.

Extends the algorithm to handle rotated rectangles with the same approximation ratio.

Breaks the 3/2 approximation barrier for the problem with rotations.

Abstract

We study the strip packing problem, a classical packing problem which generalizes both bin packing and makespan minimization. Here we are given a set of axis-parallel rectangles in the two-dimensional plane and the goal is to pack them in a vertical strip of a fixed width such that the height of the obtained packing is minimized. The packing must be non-overlapping and the rectangles cannot be rotated. A reduction from the partition problem shows that no approximation better than 3/2 is possible for strip packing in polynomial time (assuming P$\neq$NP). Nadiradze and Wiese [SODA16] overcame this barrier by presenting a $(\frac{7}{5}+\epsilon)$-approximation algorithm in pseudo-polynomial-time (PPT). As the problem is strongly NP-hard, it does not admit an exact PPT algorithm. In this paper, we make further progress on the PPT approximability of strip packing, by presenting a $(\frac43+\epsilon)$-approximation algorithm. Our result is based on a non-trivial repacking of some rectangles in the \emph{empty space} left by the construction by Nadiradze and Wiese, and in some sense pushes their approach to its limit. Our PPT algorithm can be adapted to the case where we are allowed to rotate the rectangles by $90^\circ$, achieving the same approximation factor and breaking the polynomial-time approximation barrier of 3/2 for the case with rotations as well.