Counting and enumerating maximal and maximum independent sets are well-studied problems in graph theory. In this paper we introduce methods to count and enumerate maximal/maximum independent sets in threshold graphs and k-threshold graphs and improve former results for these problems. The results can be applied to combinatorial optimization problems, and in particular to different variations of the knapsack problem. As feasible solutions for instances of those problems correspond to independent sets in threshold graphs and k-threshold graphs, we obtain polynomial time results for special knapsack and multidimensional knapsack instances. Also, we show lower and upper bounds for the number of necessary bins in several bin packing problems.

中文翻译：

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计算和枚举独立组合以及组合优化问题的应用程序

对最大和最大独立集进行计数和枚举是图论中经过充分研究的问题。在本文中，我们介绍了对阈值图和k阈值图中的最大/最大独立集进行计数和枚举的方法，并改进了这些问题的先前结果。该结果可以应用于组合优化问题，尤其是背包问题的不同变体。由于针对这些问题的实例的可行解对应于阈值图和k阈值图中的独立集合，因此我们获得了特殊背包和多维背包实例的多项式时间结果。此外，我们在几个装箱问题中显示了必要装箱数量的上限和下限。