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

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.

💡 Research Summary

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The paper addresses the classic Strip Packing problem, where a set of axis‑parallel rectangles must be placed without overlap inside a vertical strip of fixed width W and unbounded height, minimizing the total height used. This problem is NP‑hard, and while a polynomial‑time approximation scheme (PTAS) exists, the best known absolute‑approximation algorithms are either complex or have relatively high constants. The Bottom‑Left (BL) algorithm, introduced by Baker, Coffman, and Rivest in 1980, is a simple heuristic that processes rectangles in a predetermined order, always placing each rectangle at the lowest feasible position and, in case of ties, the leftmost one. The performance of BL depends critically on the ordering of the rectangles. For the ordering by decreasing width, BL is known to achieve an absolute approximation ratio of 3, and no better ordering has been discovered for more than four decades.

The authors propose a novel ordering, called the F Q W ordering, which yields a provably better bound of 13/6 ≈ 2.167. The construction proceeds in three steps:

1. F‑set (Tall rectangles) – Scan the rectangles in decreasing height and greedily add each rectangle to F as long as the total width of F remains ≤ W. Thus F contains the tallest rectangles that can be placed side‑by‑side on the bottom of the strip.

2. W‑set (Wide rectangles) – From the remaining rectangles, collect those whose width exceeds W/2. These rectangles are “wide” and can occupy at most one column each.

3. Q‑set (Remaining rectangles) – All other rectangles belong to Q.

The algorithm first packs all rectangles of F, then all of W (sorted by decreasing width), and finally the rectangles of Q in any order. This three‑phase schedule is the essence of the F Q W ordering.

To analyze the resulting packing, the authors introduce a horizontal strip partition. For each rectangle r_i placed by BL, they define a region H_i as the horizontal slab between the bottom of r_i and the highest bottom among the previously placed rectangles. Within each H_i they study proper horizontal lines (lines that do not intersect rectangle tops or bottoms) and the pattern of occupied versus unoccupied gaps. Two lemmas are central:

• Lemma 2 shows that any unoccupied gap on a proper line inside H_i must be strictly narrower than w_i, otherwise BL would have placed r_i lower.

• Lemma 3 proves that if a proper line intersects k rectangles whose widths are at least w_i, then at least k/(2k+1) of the line’s length is occupied (≥ 1/2 if the line’s leftmost or rightmost point is already occupied).

When rectangles are ordered by decreasing width, every region satisfies Lemma 3, leading to the classic 3‑approximation proof. In the F Q W ordering, most regions still satisfy the ≥ 1/2 occupancy guarantee, but regions that contain a wide rectangle from W may have a slightly lower occupancy (about 0.45). To handle these “special” regions, the authors formulate a quadratic program that captures the worst‑case distribution of occupied and unoccupied gaps under the constraints of Lemma 2 and Lemma 3. Solving this program yields an upper bound on the total unoccupied area across all special regions.

Combining the area bound for regular regions (≥ 1/2 occupied) with the quadratic‑program bound for special regions, the authors prove that the total area occupied by the BL packing is at least (6/13) · W·h_OPT. Consequently, the height of the BL packing satisfies

 h_BL ≤ (13/6) · h_OPT.

Thus the Bottom‑Left algorithm, when supplied with the F Q W ordering, achieves an absolute approximation ratio of 13/6, improving the long‑standing bound of 3. This result narrows the gap between the known lower bound of 4/3 − ε and the previous upper bound of 3, and suggests that the simple BL heuristic may eventually rival more sophisticated algorithms such as the current best 5/3 + ε approximation.

Beyond the immediate improvement, the paper’s techniques—particularly the horizontal strip partition, the occupancy lemmas, and the quadratic‑program analysis—are likely to be useful in related settings, including multiple‑strip packing, online variants, and problems where rotations are allowed. The work also reinforces the perspective that careful pre‑ordering can dramatically enhance the performance of fixed‑priority heuristics, a principle that may inspire further research on ordering strategies for other geometric optimization problems.