The Bottom-Left Algorithm for the Strip Packing Problem

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.

Figures (6)

• Figure 1: The two optimum solutions for packing the rectangles $(3,2)$, $(3,2)$, $(2,1)$, $(2,1)$, $(2,1)$, $(2,1)$, $(1,1)$ into a strip of width 7.
• Figure 2: (a) The bottom-left packing of $\mathcal{I}_{0.2}$ resulting from the optimum solution shown in the left of \ref{['fig:opt_packing_4/3']}. (b) The bottom-left packing if the rectangle of size $(1,1+\varepsilon)$ is placed before the two top rectangles of size $(2,1)$ resulting from the optimum solution shown in the right of \ref{['fig:opt_packing_4/3']}. (c) The best possible bottom-left packing if at least one of the two top rectangles of size $(2,1)$ is placed before the rectangle of size $(1,1+\varepsilon)$.
• Figure 3: (a) and (b) show the two optimal packings for the squares of sizes 3, 3, 2, 2, 2, 2, 1 in a strip of width 7. (c) shows the best bottom-left packing for the modified square packing instance ${\cal I}_{0.1}$.
• Figure 4: The unique optimal square packing up to symmetry.
• Figure 5: (a) shows that if the square of size $h$ is above two rows of squares of size $h+1$, then there is no other square that can fill the red area. (b) shows that if the square of size $h$ is between squares of size $h+1$, then there is no other square that can fill the red area. The only possibility where the square of size $h$ is between squares of size $h+1$ is (up to symmetry) shown in (c). Up to symmetry the structure shown in (d) must be part of the optimal packing.

Theorems & Definitions (12)

• theorem thmcountertheorem
• theorem thmcountertheorem
• corollary thmcountercorollary
• theorem thmcountertheorem
• theorem thmcountertheorem
• proof
• corollary thmcountercorollary
• proof
• theorem thmcountertheorem
• proof