The Bottom-Left Algorithm for the Strip Packing Problem

TL;DR

This paper analyzes the bottom-left heuristic for strip packing, improving lower bounds on its approximation ratio, especially for squares, and explores the complexity of local search algorithms based on it.

Contribution

It establishes new lower bounds for the approximation ratio of the bottom-left algorithm, including for squares, and investigates the complexity of local search variants.

Findings

Lower bound for general case improved to 4/3.

Lower bound for squares improved to 12/11.

Local search algorithms can have exponential complexity.

Abstract

The bottom-left algorithm is a simple heuristic for the Strip Packing Problem. It places the rectangles in the given order at the lowest free position in the strip, using the left most position in case of ties. Despite its simplicity, the exact approximation ratio of the bottom-left algorithm remains unknown. We will improve the more-than-40-year-old value for the lower bound from $5/4$ to $4/3 - \varepsilon$. Additionally, we will show that this lower bound holds even in the special case of squares, where the previously known lower bound was $12/11 -\varepsilon$. These lower bounds apply regardless of the ordering of the rectangles. When squares are arranged in the worst possible order, we establish a constant upper bound and a $10/3-\varepsilon$ lower bound for the approximation ratio of the bottom-left algorithm. This bound also applies to some online setting and yields an almost tight result there. Finally, we show that the approximation ratio of a local search algorithm based on permuting rectangles in the ordering of the bottom-left algorithm is at least~$2$ and that such an algorithm may need an exponential number of improvement steps to reach a local optimum.