聪明的购物者问题
The Clever Shopper Problem
摘要：
我们研究了Blazewicz等人（2010）提出的所谓“网络购物问题”的一个变体，即顾客希望以最低的总成本从商店购买一系列产品，当购买超过一定阈值时，商店会提供折扣。虽然这个问题是NP-难的，我们提供了几种情况下的精确算法，例如，当每个商店只卖两个项目，和FPT算法的项目数量，或商店的数量时，所有的价格是平等的。我们补充每个结果的硬度证明，以绘制一个严格的边界之间的听话和棘手的情况下。最后，我们给出了最大折扣和问题的一个近似算法和一些硬结果。
英文摘要：
We investigate a variant of the so-called "Internet Shopping Problem" introduced by Blazewicz et al. (2010), where a customer wants to buy a list of products at the lowest possible total cost from shops which offer discounts when purchases exceed a certain threshold. Although the problem is NP-hard, we provide exact algorithms for several cases, e.g. when each shop sells only two items, and an FPT algorithm for the number of items, or for the number of shops when all prices are equal. We complement each result with hardness proofs in order to draw a tight boundary between tractable and intractable cases. Finally, we give an approximation algorithm and hardness results for the problem of maximising the sum of discounts.
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