[Paper Review] An Absolute 2-Approximation Algorithm for Two-Dimensional Bin Packing

Rolf Harren, Rob van Stee|ArXiv.org|Mar 13, 2009

Optimization and Packing Problems13 references3 citations

TL;DR

This paper presents a polynomial-time 2-approximation algorithm for two-dimensional bin packing without rotations, achieving an absolute worst-case ratio of 2—proven optimal unless P = NP. The algorithm uses a combination of Steinberg's strip packing method, careful item classification (wide, high, small, big), and case-based packing strategies based on item size and area bounds to ensure no more than twice the optimal number of bins are used for any input.

ABSTRACT

We consider the problem of packing rectangles into bins that are unit squares, where the goal is to minimize the number of bins used. All rectangles have to be packed non-overlapping and orthogonal, i.e., axis-parallel. We present an algorithm for this problem with an absolute worst-case ratio of 2, which is optimal provided P != NP.

Motivation & Objective

• To develop a polynomial-time approximation algorithm for two-dimensional bin packing with an absolute worst-case ratio guarantee, rather than relying on asymptotic performance.
• To close the gap between asymptotic approximation ratios (which improve with large inputs) and practical performance on small or moderate inputs.
• To provide a performance guarantee valid for all inputs, not just large ones, by focusing on the absolute approximation ratio instead of the asymptotic ratio.
• To show that a 2-approximation is optimal for 2D bin packing without rotations, assuming P ≠ NP.
• To extend known results on strip packing and 2-stage packing to the general bin packing problem with absolute performance guarantees.

Proposed method

• Classify rectangles into wide (width > 1/2), high (height > 1/2), small (both ≤ 1/2), and big (both > 1/2) items to guide packing strategy.
• Use Steinberg’s algorithm as a subroutine to pack subsets of items into bins with area constraints, ensuring feasibility via geometric bounds.
• Apply case analysis based on total width of high items and total height of wide items, using a threshold δ ∈ (ε, 1/2] to define wide and high items.
• For instances with OPT(I) ≥ k, use the asymptotic algorithm by Bansal et al. [2]; for smaller instances (1 < OPT(I) < k), enumerate large items and apply a case-based packing strategy.
• In the core algorithm, pack items into bins B₁, Bᵢ, and C₁, C₂, ..., Cₗ using area and size constraints, with iterative transfer of small items between sets to maintain feasibility.
• Use contradiction arguments based on total packed area and area bounds to prove correctness and termination, ensuring all items are packed within 2·OPT(I) bins.

Experimental results

Research questions

• RQ1 Can a 2-approximation algorithm be designed for two-dimensional bin packing without rotations that guarantees performance for all inputs, not just asymptotically?
• RQ2 Is it possible to achieve an absolute approximation ratio of 2 for 2D bin packing, and is this ratio optimal under standard complexity assumptions?
• RQ3 How can Steinberg’s strip packing algorithm be adapted and combined with item classification to ensure absolute performance guarantees in bin packing?
• RQ4 What structural properties of item sets (e.g., total width of high items, total height of wide items) can be used to design a case-based packing strategy with bounded performance?
• RQ5 Can a polynomial-time algorithm be constructed that achieves a worst-case ratio of 2 for all inputs, even when the optimal number of bins is small?

Key findings

• The proposed algorithm achieves an absolute worst-case approximation ratio of 2, which is optimal unless P = NP.
• The algorithm runs in polynomial time and guarantees that no more than twice the optimal number of bins are used for any input instance.
• The correctness of the algorithm is proven via contradiction, showing that assuming more than 2·OPT(I) bins are needed leads to a contradiction with total area bounds.
• For instances with OPT(I) ≥ k, the asymptotic algorithm from Bansal et al. [2] is used; for smaller instances, a case-based enumeration and packing strategy ensures the 2-approximation.
• The algorithm handles all input types—small, large, and mixed—by classifying items and using Steinberg’s method on subsets with bounded width and area.
• The result improves upon prior work by providing a guaranteed performance bound for all inputs, unlike asymptotic algorithms whose performance only improves with input size.