We consider the product knapsack problem, which is the variant of the classical 0-1 knapsack problem where the objective consists of maximizing the product of the profits of the selected items. These profits are allowed to be positive or negative. We present the first fully polynomial-time approximation scheme for the product knapsack problem, which is known to be weakly NP-hard. Moreover, we analyze the approximation quality achieved by a natural extension of the classical knapsack greedy procedure to the product knapsack problem.

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近似产品背包问题

我们考虑产品背包问题，它是经典 0-1 背包问题的变体，其目标包括最大化所选商品的利润乘积。这些利润可以是正的或负的。我们提出了产品背包问题的第一个完全多项式时间近似方案，已知它是弱NP难的。此外，我们分析了通过将经典背包贪婪过程自然扩展到产品背包问题所实现的近似质量。