Entropic Column Generation and Lehmer Branch-and-Price for Bin Packing Problem
Abstract
We introduce Lehmer branch-and-price, an exact algorithm for bin packing and cutting stock problems built on the Lehmer transform, the ratio of Mellin transforms of item weights at consecutive orders. The transform maps each bin, a finite multiset of item weights, to a smooth increasing curve, and so gives the discrete problem a continuous representation on which the tools of analysis, such as convexity, exponential families and smoothing, can act. Read along this curve, the problem yields an explicit dual solution, a monotone representation of the dominance order, a family of constructive heuristics with first fit decreasing as its limit, and an integer identity that tells when the linear relaxation can decide an instance and when only a perfect packing can improve the best solution known. Lifted from items to patterns, the transform gives entropic column generation, a new form of column generation that bounds the relaxation from both sides at every iteration without solving the pricing problem exactly, maximizes a smoothed dual at an explicit rate, and measures by an entropy how much the relaxation leaves undecided. We show that the proposed algorithm outperforms the state-of-the-art methods on the standard benchmark in both the number of instances solved and the computational time.
Related
- On Lehmer's Problem
- Engel's Interval Packing Problem in the Boolean Lattice
- Aggregated Column Generation for Two-Stage Stochastic Mixed-Integer Programs
- The Sphere Packing Problem in Dimension 4 and the Twenty-Four-Cell Conjecture
- Decomposition of Sparse Integer Programs via Nonlinear Edge Encodings and Column-and-Row Generation
BibTeX
@misc{ataei2026entropic,
title = {{Entropic Column Generation and Lehmer Branch-and-Price for Bin Packing Problem}},
author = {Masoud Ataei},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01251},
eprint = {2611.01251},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01251},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Entropic Column Generation and Lehmer Branch-and-Price for Bin Packing Problem
AU - Masoud Ataei
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01251
UR - https://arxiv.org/abs/2611.01251
AB - We introduce Lehmer branch-and-price, an exact algorithm for bin packing and cutting stock problems built on the Lehmer transform, the ratio of Mellin transforms of item weights at consecutive orders. The transform maps each bin, a finite multiset of item weights, to a smooth increasing curve, and so gives the discrete problem a continuous representation on which the tools of analysis, such as convexity, exponential families and smoothing, can act. Read along this curve, the problem yields an explicit dual solution, a monotone representation of the dominance order, a family of constructive heuristics with first fit decreasing as its limit, and an integer identity that tells when the linear relaxation can decide an instance and when only a perfect packing can improve the best solution known. Lifted from items to patterns, the transform gives entropic column generation, a new form of column generation that bounds the relaxation from both sides at every iteration without solving the pricing problem exactly, maximizes a smoothed dual at an explicit rate, and measures by an entropy how much the relaxation leaves undecided. We show that the proposed algorithm outperforms the state-of-the-art methods on the standard benchmark in both the number of instances solved and the computational time.
ER -