Computing sets of high quality solutions has gained increasing interest in recent years. In this paper, we investigate how to obtain sets of optimal solutions for the classical knapsack problem. We present an algorithm to count exactly the number of optima to a zero-one knapsack problem instance. In addition, we show how to efficiently sample uniformly at random from the set of all global optima. In our experimental study, we investigate how the number of optima develops for classical random benchmark instances dependent on their generator parameters. We find that the number of global optima can increase exponentially for practically relevant classes of instances with correlated weights and profits which poses a justification for the considered exact counting problem.

中文翻译：

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背包问题Optima的精确计数和采样

近年来，高质量解决方案的计算集越来越受到关注。在本文中，我们研究了如何获得经典背包问题的最优解集。我们提出了一种算法来精确计算零一背包问题实例的最优数。此外，我们展示了如何从所有全局最优值的集合中高效地随机均匀采样。在我们的实验研究中，我们研究了经典随机基准实例的最优值数量如何取决于其生成器参数。我们发现，对于具有相关权重和利润的实际相关实例类别，全局最优值的数量可以呈指数增长，这为考虑的精确计数问题提供了理由。