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.
翻译:有关 0-1 knapsack 问题的十年研究导致非常高效的算法,能够迅速解决大问题到最佳程度。这促使研究人员也调查是否存在相对较小的问题案例,这些案例对于现有解决者来说是很难解决的,并调查其难度特征。以前,作者们提出了一个新的类别,硬性0-1 knapsack 问题案例,并表明所谓的包容式最大解决方案(IMS)的特性可以成为这一类的重要硬度指标。在本文中,我们制定了几个与任意的0-1 knapsack 问题案例的国际监测系统相关的具有挑战性的问题。根据以往工作的一般特征和关于IMS 问题的新结构结果,我们为解决这些问题制定了多式和假极性时间算法。我们从中得出了一套14种计算成本昂贵的功能,我们在大约540 CPU- hours超级计算机上计算出两个大型数据集的特性。我们所拟议的特征包含与问题的经验性硬性实例有关的重要实例。在早期的文献中缺少的。在0- 1 空间模型的简单性特征和新的结构中,我们用简单的空间模型来精确地分析各种硬性模型来精确地分析。