We consider the bin packing problem with d different item sizes s i and item multiplicities a i , where all numbers are given in binary encoding. This problem formulation is also known as the one-dimensional cutting stock problem . In this work, we provide an algorithm that, for constant d , solves bin packing in polynomial time. This was an open problem for all d\ge 3 . In fact, for constant d our algorithm solves the following problem in polynomial time: Given two d -dimensional polytopes P and Q , find the smallest number of integer points in P whose sum lies in Q . Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times, and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant.

中文翻译：

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具有恒定项目类型的装箱的多项式

我们考虑装箱问题d不同的物品尺寸s 一世 和项目的多重性一种 一世 ，其中所有数字都以二进制编码形式给出。这个问题的表述也被称为一维切料问题. 在这项工作中，我们提供了一种算法，对于常数d，解决多项式时间内的装箱问题。这对所有人来说都是一个悬而未决的问题d\ge 3. 事实上，对于常数d我们的算法在多项式时间内解决了以下问题：给定两个d维多面体磷和问, 找到最小的整数点数磷谁的总和在问. 我们的方法也适用于高多样性调度问题，其中每种作业类型的副本数量以二进制编码给出，每种类型都带有某些参数，例如发布日期、处理时间和截止日期。我们表明，如果作业类型的数量是恒定的，则可以在多项式时间内解决各种高多重性调度问题。