Abstract
In this paper, we study the online knapsack problem with removal cost. The input is a sequence of items u 1,u 2,…,u n , each of which has a size and a value, where the value of each item is assumed to be equal to the size. Given the ith item u i , we either put u i into the knapsack or reject it with no cost. When u i is put into the knapsack, some items in the knapsack are removed with removal cost if the sum of the size of u i and the total size in the current knapsack exceeds the capacity of the knapsack. Here the removal cost means a cancellation charge or disposal fee. Our goal is to maximize the profit, i.e., the sum of the values of items in the last knapsack minus the total removal cost occurred.
In this paper, we consider two kinds of removal cost: unit and proportional cost. For both models, we provide their competitive ratios. Namely, we construct optimal online algorithms and prove that they are best possible.
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References
Ashwinkumar, B.V.: Buyback Problem - Approximate Matroid Intersection with Cancellation Costs. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 379–390. Springer, Heidelberg (2011)
Ashwinkumar, B.V., Kleinberg, R.: Randomized Online Algorithms for the Buyback Problem. In: Leonardi, S. (ed.) WINE 2009. LNCS, vol. 5929, pp. 529–536. Springer, Heidelberg (2009)
Babaioff, M., Hartline, J.D., Kleinberg, R.D.: Selling ad campaigns: Online algorithms with cancellations. In: ACM Conference on Electronic Commerce, pp. 61–70 (2009)
Biyalogorsky, E., Carmon, Z., Fruchter, G.E., Gerstner, E.: Research note: Overselling with opportunistic cancellations. Marketing Science 18(4), 605–610 (1999)
Constantin, F., Feldman, J., Muthukrishnan, S., Pál, M.: An online mechanism for ad slot reservations with cancellations. In: SODA, pp. 1265–1274 (2009)
Han, X., Makino, K.: Online Minimization Knapsack Problem. In: Bampis, E., Jansen, K. (eds.) WAOA 2009. LNCS, vol. 5893, pp. 182–193. Springer, Heidelberg (2010)
Han, X., Makino, K.: Online removable knapsack with limited cuts. Theoretical Computer Science 411, 3956–3964 (2010)
Iwama, K., Taketomi, S.: Removable online knapsack problems. LNCS, pp. 293–305 (2002)
Iwama, K., Zhang, G.: Optimal Resource Augmentations for Online Knapsack. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) RANDOM 2007 and APPROX 2007. LNCS, vol. 4627, pp. 180–188. Springer, Heidelberg (2007)
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer (2004)
Lueker, G.S.: Average-case analysis of off-line and on-line knapsack problems. Journal of Algorithms 29(2), 277–305 (1998)
Marchetti-Spaccamela, A., Vercellis, C.: Stochastic on-line knapsack problems. Mathematical Programming 68, 73–104 (1995)
Noga, J., Sarbua, V.: An online partially fractional knapsack problem. In: ISPAN, pp. 108–112 (2005)
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Han, X., Kawase, Y., Makino, K. (2012). Online Knapsack Problem with Removal Cost. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32241-9_6
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DOI: https://doi.org/10.1007/978-3-642-32241-9_6
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