A 13/6-Approximation for Strip Packing via the Bottom-Left Algorithm

TL;DR

The paper advances the study of Strip Packing by showing that the Bottom-Left algorithm, when rectangles are ordered by a novel FQW partition (tall-first bottom row F, wide W, and the rest Q), achieves a $13/6$-approximation. It introduces a horizontal strip partition and a detailed occupancy analysis of proper horizontal lines, culminating in a quadratic-program bound that tightly controls unoccupied space. This yields the first improvement over the classic BL bound of 3 and narrows the gap toward the lower bound near $4/3$, while also outlining the limits of BL under various natural orderings. The techniques, especially the quadratic-program framework, may be adaptable to related packing problems and variants of the BL algorithm.

Abstract

In the Strip Packing problem, we are given a vertical strip of fixed width and unbounded height, along with a set of axis-parallel rectangles. The task is to place all rectangles within the strip, without overlaps, while minimizing the height of the packing. This problem is known to be NP-hard. The Bottom-Left Algorithm is a simple and widely used heuristic for Strip Packing. Given a fixed order of the rectangles, it places them one by one, always choosing the lowest feasible position in the strip and, in case of ties, the leftmost one. Baker, Coffman, and Rivest proved in 1980 that the Bottom-Left Algorithm has approximation ratio 3 if the rectangles are sorted by decreasing width. For the past 45 years, no alternative ordering has been found that improves this bound. We introduce a new rectangle ordering and show that with this ordering the Bottom-Left Algorithm achieves a 13/6 approximation for the Strip Packing problem.

Figures (4)

• Figure 1: The packing $\text{BL}(\mathcal{I}_\mathcal{FQW})$ together with the horizontal strip partition. Blue rectangles are in $\mathcal{F}$, brown in $\mathcal{Q}$, and green in $\mathcal{W}$.
• Figure 2: Example showing that the lower horizontal line $\ell$ of order $2$ is at least half occupied. Blue rectangles are in $\mathcal{F}$ and brown ones are in $\mathcal{Q}$.
• Figure 3: Example of a horizontal line $\ell$ of order $2$. To the right of $x_\ell$ the line $\ell$ is exactly half occupied prior to placing the first rectangle above $\ell$, the remaining gap between $x_{\mathcal{F}}^\ell$ and $x_\ell$ is defined as $\beta^\ell$.
• Figure 4: The BL packing of an instance following the $\mathcal{FQW}$-ordering on a strip of width $W=3w$ with rectangles $\mathcal{F} = \{(w,h+1),(w+1,h) \}$, $\mathcal{Q} = \{(w+1,h),(w+1,1) (w+1,h)\}$ and $\mathcal{W} = \varnothing$.

Theorems & Definitions (43)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6