基于包含极大解的0-1背包问题的特征

Jorik Jooken, Pieter Leyman, P. D. Causmaecker
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引用次数: 0

摘要

几十年来对0-1背包问题的研究产生了非常高效的算法,能够快速解决大型问题实例并达到最优。这促使研究人员也调查是否存在相对较小的问题实例,这些问题实例对于现有的求解者来说是困难的,并调查它们的硬度特征。在此之前,作者提出了一类新的硬0-1背包问题实例,并证明了所谓的包含极大解(IMSs)的性质可以作为这类问题的重要硬度指标。在本文中,我们提出了与任意0-1背包问题实例的IMSs相关的几个新的具有计算挑战性的问题。在此基础上,我们提出了多项式和伪多项式时间算法来解决这些问题。从这里我们得到了一组14个计算上昂贵的特征,我们在一台超级计算机上用大约540 cpu小时计算两个大型数据集。我们表明,通过训练机器学习模型,所提出的特征包含与问题实例的经验硬度相关的重要信息,这些信息在文献中的早期特征中是缺失的,这些模型可以准确地预测各种0-1背包问题实例的经验硬度。使用实例空间分析方法,我们还证明了硬0-1背包问题实例聚集在实例空间的一个相对密集的区域周围,并且在实例空间的容易部分和困难部分的几个特征表现不同。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Features for the 0-1 knapsack problem based on inclusionwise maximal solutions
Decades of research on the 0-1 knapsack problem led to very efficient algorithms that are able to quickly solve large problem instances to optimality. This prompted researchers to also investigate whether relatively small problem instances exist that are hard for existing solvers and investigate which features characterize their hardness. Previously the authors proposed a new class of hard 0-1 knapsack problem instances and demonstrated that the properties of so-called inclusionwise maximal solutions (IMSs) can be important hardness indicators for this class. In the current paper, we formulate several new computationally challenging problems related to the IMSs of arbitrary 0-1 knapsack problem instances. Based on generalizations of previous work and new structural results about IMSs, we formulate polynomial and pseudopolynomial time algorithms for solving these problems. From this we derive a set of 14 computationally expensive features, which we calculate for two large datasets on a supercomputer in approximately 540 CPU-hours. We show that the proposed features contain important information related to the empirical hardness of a problem instance that was missing in earlier features from the literature by training machine learning models that can accurately predict the empirical hardness of a wide variety of 0-1 knapsack problem instances. Using the instance space analysis methodology, we also show that hard 0-1 knapsack problem instances are clustered together around a relatively dense region of the instance space and several features behave differently in the easy and hard parts of the instance space.
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