Abstract
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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References
Assmann, S., Johnson, D., Kleitman, D., Leung, J.-T.: On a dual version of the one-dimensional bin packing problem. J. Algorithms 5, 502–525 (1984)
Berman, P., Karpinski, M., Scott, A.D.: Approximation hardness of short symmetric instances of MAX-3SAT. In: Electronic Colloquium on Computational Complexity (ECCC) (2003)
Blazewicz, J., Kovalyov, M.Y., Musial, J., Urbanski, A.P., Wojciechowski, A.: Internet shopping optimization problem. Appl. Math. Comput. Sci. 20, 385–390 (2010)
Blazewicz, J., Bouvry, P., Kovalyov, M.Y., Musial, J.: Internet shopping with price sensitive discounts. 4OR 12, 35–48 (2014)
Blazewicz, J., Cheriere, N., Dutot, P.-F., Musial, J., Trystram, D.: Novel dual discounting functions for the internet shopping optimization problem: new algorithms. J. Sched. 19, 245–255 (2016)
Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Kernelization lower bounds by cross-composition. SIAM J. Discret. Math. 28, 277–305 (2014)
Cesati, M.: Perfect code is W[1]-complete. Inf. Process. Lett. 81, 163–168 (2002)
Edmonds, J.: Paths, trees and flowers. Canad. J. Math. 449–467 (1965)
Gabow, H.N.: A note on degree-constrained star subgraphs of bipartite graphs. Inf. Process. Lett. 5, 165–167 (1976)
Jansen, K., Kratsch, S., Marx, D., Schlotter, I.: Bin packing with fixed number of bins revisited. J. Comput. Syst. Sci. 79, 39–49 (2013)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Proceedings of a Symposium on the Complexity of Computer Computations. The IBM Research Symposia Series, pp. 85–103. Plenum Press, Yorktown Heights (1972)
van Bevern, R., Komusiewicz, C., Niedermeier, R., Sorge, M., Walsh, T.: H-index manipulation by merging articles: models, theory, and experiments. Artif. Intell. 240, 19–35 (2016)
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Bulteau, L., Hermelin, D., Labarre, A., Vialette, S. (2018). The Clever Shopper Problem. In: Fomin, F., Podolskii, V. (eds) Computer Science – Theory and Applications. CSR 2018. Lecture Notes in Computer Science(), vol 10846. Springer, Cham. https://doi.org/10.1007/978-3-319-90530-3_6
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DOI: https://doi.org/10.1007/978-3-319-90530-3_6
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