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The clever shopper problem. (English) Zbl 1434.68192
Summary: We investigate a variant of the so-called Internet Shopping problem introduced by J. Błażewicz et al. [Int. J. Appl. Math. Comput. Sci. 20, No. 2, 385–390 (2010; Zbl 1230.90207)], 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.
MSC:
68Q25 Analysis of algorithms and problem complexity
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
68Q27 Parameterized complexity, tractability and kernelization
68W05 Nonnumerical algorithms
68W25 Approximation algorithms
90C27 Combinatorial optimization
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