Unit Cost Buyback Problem

Abstract

In this paper, we study the unit cost buyback problem, i.e., the buyback problem with a fixed cancellation cost for each canceled element. The input of the problem is a sequence of elements e1, e2, . . . , en where each element ei has a weight w(ei). We assume that the weights are in a known range [l, u], i.e., l ≤ w(ei) ≤ u for any i. Given the i th element ei, we either accept ei or reject it with no cost, where we can keep a set of elements that satisfies a certain constraint. In order to accept a new element ei, we can cancel some previously selected elements at a cost which is proportional to the number of elements canceled. Our goal is to maximize the profit, i.e., the total 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.