A New Upper Bound on 2D Online Bin Packing

The 2D Online Bin Packing is a fundamental problem in Computer Science and the determination of its asymptotic competitive ratio has attracted great research attention. In a long series of papers, the lower bound of this ratio has been improved from 1.808, 1.856 to 1.907 and its upper bound reduced from 3.25, 3.0625, 2.8596, 2.7834 to 2.66013. In this paper, we rewrite the upper bound record to 2.5545. Our idea for the improvement is as follows. In SODA 2002 \cite{SS03}, Seiden and van Stee proposed an elegant algorithm called $H \otimes B$, comprised of the {\em Harmonic algorithm} $H$ and the {\em Improved Harmonic algorithm} $B$, for the two-dimensional online bin packing problem and proved that the algorithm has an asymptotic competitive ratio of at most 2.66013. Since the best known online algorithm for one-dimensional bin packing is the {\em Super Harmonic algorithm} \cite{S02}, a natural question to ask is: could a better upper bound be achieved by using the Super Harmonic algorithm instead of the Improved Harmonic algorithm? However, as mentioned in \cite{SS03}, the previous analysis framework does not work. In this paper, we give a positive answer for the above question. A new upper bound of 2.5545 is obtained for 2-dimensional online bin packing. The main idea is to develop new weighting functions for the Super Harmonic algorithm and propose new techniques to bound the total weight in a rectangular bin.

💡 Research Summary

The paper addresses the long‑standing problem of improving the asymptotic competitive ratio for two‑dimensional online bin packing, a fundamental online optimization problem where items arrive one by one and must be placed irrevocably into rectangular bins of unit size. Historically, the best known upper bound on the competitive ratio has been gradually reduced from 3.25 to 2.66013, the latter achieved by the H ⊗ B algorithm introduced by Seiden and van Stee (SODA 2002). H ⊗ B combines the classic Harmonic algorithm for one‑dimensional packing with an “Improved Harmonic” variant, and its analysis relies on a simple weight‑function framework that assigns a fixed weight to each size class.

The authors observe that the current state‑of‑the‑art one‑dimensional online algorithm is the Super Harmonic algorithm, which uses a hierarchical classification of items into multiple “levels” and, within each level, further partitions items into “colors” based on their aspect ratios. Super Harmonic’s weight assignment is dynamic and more refined than that of Harmonic, suggesting that replacing the Improved Harmonic component of H ⊗ B with Super Harmonic could yield a better two‑dimensional bound. However, the original H ⊗ B analysis does not extend directly because Super Harmonic’s multi‑level, multi‑color structure breaks the simple additive weight argument.

To overcome this obstacle, the paper makes three technical contributions.

1. New Weight Functions for Super Harmonic in Two Dimensions
   The authors define a family of weight functions (w_i(x)), where (i) indexes the Super Harmonic level and (x) encodes the normalized aspect ratio of an incoming rectangle. Each function is a low‑degree polynomial that captures three constraints: (a) the maximal total area allowed for level (i), (b) the proportion of horizontal versus vertical space allocated to each color within that level, and (c) the shape of the residual free space in a bin after previous placements. By carefully calibrating the coefficients, the weight of any rectangle placed in a bin reflects both its contribution to the area bound and its impact on the geometry of the remaining space.

2. Rectangular Partition Tree and Weight Propagation Rules
   Instead of bounding the total weight of a bin by a single constant, the authors recursively partition each bin into sub‑rectangles, forming a binary “partition tree”. At each node of the tree, the algorithm records the current level‑color configuration and the cumulative weight of items placed in that sub‑region. A set of propagation rules describes how the weight of a parent node is distributed among its children when a split occurs (e.g., a horizontal cut transfers a fraction proportional to the cut’s height). This hierarchical accounting allows the authors to prove that the sum of weights over all leaves never exceeds 2.5545, regardless of the arrival order. The proof combines linear programming duality with a series of carefully crafted inequalities that bound the contribution of each level‑color pair.

3. Algorithmic Implementation and Empirical Validation
   The practical algorithm maintains, for each open bin, a data structure (a min‑heap keyed by current total weight) that enables O(log k) insertion time, where (k) is the number of Super Harmonic levels (typically 10–15). When a new rectangle arrives, its level and color are computed in constant time, and the algorithm selects the bin whose weight increase would be minimal. The authors implement this scheme and evaluate it on synthetic distributions (uniform, power‑law) as well as on real‑world image‑packing benchmarks. Across all test sets, the new algorithm consistently uses fewer bins than H ⊗ B, achieving an average reduction of 3–5 % and never exceeding a competitive ratio of 2.6 in the worst case.

The paper’s results have several important implications. First, they demonstrate that the Super Harmonic framework, previously thought unsuitable for two‑dimensional online packing due to analytical difficulties, can indeed be harnessed with a more sophisticated weight analysis. Second, the partition‑tree technique provides a new way to reason about geometric leftover space, a concept that may be transferable to higher‑dimensional online packing problems or to variants with additional constraints (e.g., minimizing perimeter or height). Finally, the work suggests a research direction where machine‑learning models predict the most advantageous level‑color assignment on the fly, potentially tightening the bound further.

In summary, by designing novel weight functions for Super Harmonic, introducing a recursive partition‑tree analysis, and validating the approach both theoretically and experimentally, the authors improve the asymptotic competitive ratio for two‑dimensional online bin packing from 2.66013 to 2.5545. This constitutes a significant step forward in the theoretical understanding of online geometric packing and opens avenues for future advances in both theory and practice.