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

2025-09-04

AI Summary

Strip Packing requires packing a set of axis-aligned rectangles into a vertical strip of fixed width and unbounded height, without overlap, minimizing the total height; it is NP-hard. This paper proposes a novel rectangle ordering strategy for the classical Bottom-Left (BL) heuristic—breaking, for the first time in 45 years, its long-standing approximation ratio barrier. Under the new ordering, we prove that BL achieves an approximation ratio of 13/6 (≈2.167), strictly improving upon the prior worst-case bound of 3 attained under width-nonincreasing order. Our analysis leverages geometric structural insights and a redefinition of positional priorities, requiring no modification to the original BL placement rules—only input sequence optimization suffices for this performance leap. This yields the best-known deterministic heuristic approximation guarantee for Strip Packing to date.

Technology Category

• Search and Optimization: Heuristic Search
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Improving approximation ratio for NP-hard Strip Packing problem
• Developing new rectangle ordering for Bottom-Left Algorithm
• Achieving 13/6 approximation bound for packing optimization

Innovation

• New rectangle ordering for Bottom-Left Algorithm
• Achieves 13/6 approximation ratio
• Improves upon 45-year-old 3-approximation bound