Proportional Cost Buyback Problem with Weight Bounds
In this paper, we study the proportional cost buyback problem. The input is a sequence of elements \(e_1,e_2,\dots ,e_n\), each of which has a weight \(w(e_i)\). We assume that weights have an upper and a lower bound, i.e., \(l\le w(e_i)\le u\) for any \(i\). Given the ith element \(e_i\), we either accept \(e_i\) or reject it with no cost, subject to some constraint on the set of accepted elements. During the iterations, we could cancel some previously accepted elements at a cost that is proportional to the total weight of them. Our goal is to maximize the profit, i.e., the sum of the weights of elements kept until the end minus the total cancellation cost occurred. We consider the matroid and unweighted knapsack constraints. For either case, we construct optimal online algorithms and prove that they are the best possible.
KeywordsKnapsack Problem Competitive Ratio Element Weight Knapsack Constraint Weight Bound
The first author is supported by JSPS KAKENHI Grant Number 26887014 and JST, ERATO, Kawarabayashi Large Graph Project. The second author is supported by NFSC(11571060), RGC (HKU716412E) and the Fundamental Research Funds for the Central Universities (DUT15LK10). The last author is supported by JSPS KAKENHI Grant.
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