Unit cost buyback problem

Abstract

In this paper, we study unit cost buyback problem, i.e., the buyback problem with fixed cancellation cost for each cancelled item. The input is a sequence of elements e1,e2,...,en, each of which has a weight w(ei). We assume that weights have an upper and a lower bound, i.e., l ≤ w(ei) ≤ u for any i. Given the ith element ei, we either accept ei or reject it with no cost, subject to some constraint on the set of accepted elements. In order to accept a new element ei, we could cancel some previous selected elements at a cost which is proportional to the number of elements canceled. Our goal is to maximize the profit, i.e., the sum of the weights of elements accepted (and not canceled) minus the total cancellation cost occurred. We construct optimal online algorithms and prove that they are the best possible, when the constraint is a matroid constraint or the unweighted knapsack constraint. © 2013 Springer-Verlag.