← 研踪 PaperTrailarXiv:2611.01251 · v1
Entropic Column Generation and Lehmer Branch-and-Price for Bin Packing Problem
首次提交 2026-11-01 04:32
版本更新 2026-11-01 04:32
雷达观测 2026-11-04 23:43
摘要
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.
作者备注：38 pages, 1 figure, 8 tables. Code: https://github.com/chimera1001/lehmer-bp
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v1 · 2026-11-01 04:32 · 最新观测版本
Entropic Column Generation and Lehmer Branch-and-Price for Bin Packing Problem
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.
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