Unit Cost Buyback Problem
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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.
KeywordsOnline algorithm Competitive analysis Buyback problem Matroid Knapsack problem
The first author was supported by JST, ERATO, Kawarabayashi Large Graph Project and JSPS KAKENHI Grant Numbers 26887014 and 16K16005. The second author was supported by RGC (HKU716412E) and NSFC (11571060). The last author was supported by JSPS KAKENHI Grant Number JP24106002, JP25280004, JP26280001, and JST CREST Grant Number JPMJCR1402.
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